Semantic Graph — Visual Examination Report

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\mu = \\sum_{i=1}^{n} x_i p_i",
      "chartScript": {
        "script": "mu - n*p_i*x_i",
        "variables": [
          "mu",
          "n",
          "p_i",
          "x_i"
        ]
      }
    },
    {
      "id": "mu",
      "type": "scalar",
      "latex": "\\mu",
      "subexpr": "\\mu",
      "chartScript": {
        "script": "mu",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "i",
      "type": "scalar",
      "latex": "i",
      "subexpr": "i",
      "chartScript": {
        "script": "i",
        "variables": [
          "i"
        ]
      }
    },
    {
      "id": "__num_3",
      "type": "number",
      "label": "1",
      "subexpr": "1",
      "chartScript": {
        "script": "1",
        "variables": []
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "__equals_4",
      "type": "relation",
      "op": "equals",
      "subexpr": "i = 1",
      "chartScript": {
        "script": "i - 1",
        "variables": [
          "i"
        ]
      }
    },
    {
      "id": "__sum_2",
      "type": "operator",
      "op": "sum",
      "with_respect_to": "i",
      "lower_bound": "__num_3",
      "upper_bound": "n",
      "subexpr": "\\sum_{i=1}^{n} p_{i} x_{i}",
      "chartScript": {
        "script": "n*p_i*x_i",
        "variables": [
          "n",
          "p_i",
          "x_i"
        ]
      }
    },
    {
      "id": "__multiply_5",
      "type": "operator",
      "op": "multiply",
      "subexpr": "x_{i} p_{i}",
      "chartScript": {
        "script": "p_i*x_i",
        "variables": [
          "p_i",
          "x_i"
        ]
      }
    },
    {
      "id": "x_i",
      "type": "scalar",
      "latex": "x_{i}",
      "subexpr": "x_{i}",
      "chartScript": {
        "script": "x_i",
        "variables": [
          "x_i"
        ]
      }
    },
    {
      "id": "p_i",
      "type": "scalar",
      "latex": "p_{i}",
      "subexpr": "p_{i}",
      "chartScript": {
        "script": "p_i",
        "variables": [
          "p_i"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "mu",
      "to": "__equals_1"
    },
    {
      "from": "i",
      "to": "__sum_2",
      "role": "wrt"
    },
    {
      "from": "i",
      "to": "__equals_4"
    },
    {
      "from": "__num_3",
      "to": "__equals_4"
    },
    {
      "from": "__equals_4",
      "to": "__sum_2",
      "role": "lb"
    },
    {
      "from": "n",
      "to": "__sum_2",
      "role": "ub"
    },
    {
      "from": "x_i",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "p_i",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_5",
      "to": "__sum_2"
    },
    {
      "from": "__sum_2",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  mu["$\mu$"]:::scalar
  i["$i$"]:::scalar
  __num_3["$1$"]:::number
  n["$n$"]:::scalar
  __equals_4{"="}:::relation
  __sum_2(("$\sum_{i}$")):::operator
  __multiply_5(("$\times$")):::operator
  x_i["$x_{i}$"]:::scalar
  p_i["$p_{i}$"]:::scalar
  mu --> __equals_1
  i -->|wrt| __sum_2
  i --> __equals_4
  __num_3 --> __equals_4
  __equals_4 -->|lb| __sum_2
  n -->|ub| __sum_2
  x_i --> __multiply_5
  p_i --> __multiply_5
  __multiply_5 --> __sum_2
  __sum_2 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\sigma^2 = E(X^2) - \\mu^2",
      "chartScript": {
        "script": "pow(mu, 2) + pow(sigma, 2) - E(X**2)",
        "variables": [
          "X",
          "mu",
          "sigma"
        ]
      }
    },
    {
      "id": "__power_2",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "\\sigma^{2}",
      "chartScript": {
        "script": "pow(sigma, 2)",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "sigma",
      "type": "scalar",
      "latex": "\\sigma",
      "subexpr": "\\sigma",
      "chartScript": {
        "script": "sigma",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "__add_3",
      "type": "operator",
      "op": "add",
      "subexpr": "-\\mu^{2} + E{\\left(X^{2} \\right)}",
      "chartScript": {
        "script": "E*pow(X, 2) - pow(mu, 2)",
        "variables": [
          "E",
          "X",
          "mu"
        ]
      }
    },
    {
      "id": "__E_4",
      "type": "function",
      "op": "E",
      "subexpr": "E{\\left(X^{2} \\right)}",
      "chartScript": {
        "script": "E*pow(X, 2)",
        "variables": [
          "E",
          "X"
        ]
      }
    },
    {
      "id": "__power_5",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "X^{2}",
      "chartScript": {
        "script": "pow(X, 2)",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "__negation_6",
      "type": "operator",
      "op": "negation",
      "subexpr": "-\\mu^{2}",
      "chartScript": {
        "script": "-pow(mu, 2)",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "__power_7",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "\\mu^{2}",
      "chartScript": {
        "script": "pow(mu, 2)",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "mu",
      "type": "scalar",
      "latex": "\\mu",
      "subexpr": "\\mu",
      "chartScript": {
        "script": "mu",
        "variables": [
          "mu"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "sigma",
      "to": "__power_2"
    },
    {
      "from": "__power_2",
      "to": "__equals_1"
    },
    {
      "from": "X",
      "to": "__power_5"
    },
    {
      "from": "__power_5",
      "to": "__E_4"
    },
    {
      "from": "__E_4",
      "to": "__add_3"
    },
    {
      "from": "mu",
      "to": "__power_7"
    },
    {
      "from": "__power_7",
      "to": "__negation_6"
    },
    {
      "from": "__negation_6",
      "to": "__add_3"
    },
    {
      "from": "__add_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __power_2(("${(\cdot)}^{2}$")):::operator
  sigma["$\sigma$"]:::scalar
  __add_3(("$+$")):::operator
  __E_4{{"$E(\cdot)$"}}:::function
  __power_5(("${(\cdot)}^{2}$")):::operator
  X["$X$"]:::scalar
  __negation_6@{ shape: "flip-tri", label: "$-$" }
  __power_7(("${(\cdot)}^{2}$")):::operator
  mu["$\mu$"]:::scalar
  class __negation_6 operator
  sigma --> __power_2
  __power_2 --> __equals_1
  X --> __power_5
  __power_5 --> __E_4
  __E_4 --> __add_3
  mu --> __power_7
  __power_7 --> __negation_6
  __negation_6 --> __add_3
  __add_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#ef5350,stroke-width:6px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#ef5350,stroke-width:6px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#ef5350,stroke-width:6px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "P(A \\mid B) = \\frac{P(B \\mid A) P(A)}{P(B)}",
      "chartScript": {
        "script": "-P(A)*P(A*B*mid)/P(B) + P(A*B*mid)",
        "variables": [
          "A",
          "B",
          "mid"
        ]
      }
    },
    {
      "id": "__P_2",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\mid B \\right)}",
      "chartScript": {
        "script": "A*B*P*mid",
        "variables": [
          "A",
          "B",
          "P",
          "mid"
        ]
      }
    },
    {
      "id": "A",
      "type": "scalar",
      "latex": "A",
      "subexpr": "A",
      "chartScript": {
        "script": "A",
        "variables": [
          "A"
        ]
      }
    },
    {
      "id": "B",
      "type": "scalar",
      "latex": "B",
      "subexpr": "B",
      "chartScript": {
        "script": "B",
        "variables": [
          "B"
        ]
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "P{\\left(B \\mid A \\right)} P{\\left(A \\right)} \\frac{1}{P{\\left(B \\right)}}",
      "chartScript": {
        "script": "pow(A, 2)*P*mid",
        "variables": [
          "A",
          "P",
          "mid"
        ]
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "P{\\left(B \\mid A \\right)} P{\\left(A \\right)}",
      "chartScript": {
        "script": "pow(A, 2)*B*pow(P, 2)*mid",
        "variables": [
          "A",
          "B",
          "P",
          "mid"
        ]
      }
    },
    {
      "id": "__P_5",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(B \\mid A \\right)}",
      "chartScript": {
        "script": "A*B*P*mid",
        "variables": [
          "A",
          "B",
          "P",
          "mid"
        ]
      }
    },
    {
      "id": "__P_6",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\right)}",
      "chartScript": {
        "script": "A*P",
        "variables": [
          "A",
          "P"
        ]
      }
    },
    {
      "id": "__power_7",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{P{\\left(B \\right)}}",
      "chartScript": {
        "script": "1/(B*P)",
        "variables": [
          "B",
          "P"
        ]
      }
    },
    {
      "id": "__P_8",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(B \\right)}",
      "chartScript": {
        "script": "B*P",
        "variables": [
          "B",
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "A",
      "to": "__P_2"
    },
    {
      "from": "B",
      "to": "__P_2",
      "role": "condition"
    },
    {
      "from": "__P_2",
      "to": "__equals_1"
    },
    {
      "from": "B",
      "to": "__P_5"
    },
    {
      "from": "A",
      "to": "__P_5",
      "role": "condition"
    },
    {
      "from": "__P_5",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "A",
      "to": "__P_6"
    },
    {
      "from": "__P_6",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "B",
      "to": "__P_8"
    },
    {
      "from": "__P_8",
      "to": "__power_7"
    },
    {
      "from": "__power_7",
      "to": "__multiply_3"
    },
    {
      "from": "__multiply_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __P_2{{"$P(\cdot\mid \cdot)$"}}:::function
  A["$A$"]:::scalar
  B["$B$"]:::scalar
  __multiply_3(("$\times$")):::operator
  __multiply_4(("$\times$")):::operator
  __P_5{{"$P(\cdot\mid \cdot)$"}}:::function
  __P_6{{"$P(\cdot)$"}}:::function
  __power_7(("$\dfrac{1}{(\cdot)}$")):::operator
  __P_8{{"$P(\cdot)$"}}:::function
  A --> __P_2
  B -->|condition| __P_2
  __P_2 --> __equals_1
  B --> __P_5
  A -->|condition| __P_5
  __P_5 --> __multiply_4
  A --> __P_6
  __P_6 --> __multiply_4
  __multiply_4 --> __multiply_3
  B --> __P_8
  __P_8 --> __power_7
  __power_7 -.-> __multiply_3
  __multiply_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#ef5350,stroke-width:3px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#42a5f5,stroke-width:1px
  linkStyle 12 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}",
      "chartScript": {
        "script": "f(x) - 1/2*sqrt(2)*exp(-1/2*pow(-mu + x, 2)/pow(sigma, 2))/(sqrt(pi)*sigma)",
        "variables": [
          "mu",
          "sigma",
          "x"
        ]
      }
    },
    {
      "id": "__f_2",
      "type": "function",
      "op": "f",
      "subexpr": "f{\\left(x \\right)}",
      "chartScript": {
        "script": "f*x",
        "variables": [
          "f",
          "x"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "e^{- \\frac{\\left(- \\mu + x\\right)^{2}}{2 \\sigma^{2}}} \\frac{1}{\\sigma \\sqrt{2 \\pi}}",
      "chartScript": {
        "script": "(1/2)*sqrt(2)*exp(-1/2*pow(-mu + x, 2)/pow(sigma, 2))/(sqrt(pi)*sigma)",
        "variables": [
          "mu",
          "sigma",
          "x"
        ]
      }
    },
    {
      "id": "__power_4",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{\\sigma \\sqrt{2 \\pi}}",
      "chartScript": {
        "script": "(1/2)*sqrt(2)/(sqrt(pi)*sigma)",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "__multiply_5",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\sigma \\sqrt{2 \\pi}",
      "chartScript": {
        "script": "sqrt(2)*sqrt(pi)*sigma",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "sigma",
      "type": "scalar",
      "latex": "\\sigma",
      "subexpr": "\\sigma",
      "chartScript": {
        "script": "sigma",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "__power_6",
      "type": "operator",
      "op": "power",
      "exponent": "1/2",
      "subexpr": "\\sqrt{2 \\pi}",
      "chartScript": {
        "script": "sqrt(2)*sqrt(pi)",
        "variables": []
      }
    },
    {
      "id": "__multiply_7",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\pi 2",
      "chartScript": {
        "script": "2*pi",
        "variables": []
      }
    },
    {
      "id": "__num_8",
      "type": "number",
      "label": "2",
      "subexpr": "2",
      "chartScript": {
        "script": "2",
        "variables": []
      }
    },
    {
      "id": "pi",
      "type": "constant",
      "latex": "\\pi",
      "subexpr": "\\pi",
      "chartScript": {
        "script": "pi",
        "variables": []
      }
    },
    {
      "id": "__power_9",
      "type": "operator",
      "op": "power",
      "subexpr": "e^{- \\frac{\\left(- \\mu + x\\right)^{2}}{2 \\sigma^{2}}}",
      "chartScript": {
        "script": "exp(-1/2*pow(-mu + x, 2)/pow(sigma, 2))",
        "variables": [
          "mu",
          "sigma",
          "x"
        ]
      }
    },
    {
      "id": "e",
      "type": "scalar",
      "latex": "e",
      "subexpr": "e",
      "chartScript": {
        "script": "e",
        "variables": []
      }
    },
    {
      "id": "__negation_10",
      "type": "operator",
      "op": "negation",
      "subexpr": "-\\left(- \\mu + x\\right)^{2} \\frac{1}{\\sigma^{2}} \\frac{1}{2}",
      "chartScript": {
        "script": "-1/2*pow(-mu + x, 2)/pow(sigma, 2)",
        "variables": [
          "mu",
          "sigma",
          "x"
        ]
      }
    },
    {
      "id": "__multiply_11",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\left(- \\mu + x\\right)^{2} \\frac{1}{\\sigma^{2}} \\frac{1}{2}",
      "chartScript": {
        "script": "(1/2)*pow(-mu + x, 2)/pow(sigma, 2)",
        "variables": [
          "mu",
          "sigma",
          "x"
        ]
      }
    },
    {
      "id": "__power_12",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "\\left(- \\mu + x\\right)^{2}",
      "chartScript": {
        "script": "pow(-mu + x, 2)",
        "variables": [
          "mu",
          "x"
        ]
      }
    },
    {
      "id": "__add_13",
      "type": "operator",
      "op": "add",
      "subexpr": "x - \\mu",
      "chartScript": {
        "script": "-mu + x",
        "variables": [
          "mu",
          "x"
        ]
      }
    },
    {
      "id": "__negation_14",
      "type": "operator",
      "op": "negation",
      "subexpr": "-\\mu",
      "chartScript": {
        "script": "-mu",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "mu",
      "type": "scalar",
      "latex": "\\mu",
      "subexpr": "\\mu",
      "chartScript": {
        "script": "mu",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "__multiply_15",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{1}{\\sigma^{2}} \\frac{1}{2}",
      "chartScript": {
        "script": "(1/2)/pow(sigma, 2)",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "__num_16",
      "type": "number",
      "label": "1/2",
      "subexpr": "\\frac{1}{2}",
      "chartScript": {
        "script": "1/2",
        "variables": []
      }
    },
    {
      "id": "__power_17",
      "type": "operator",
      "op": "power",
      "exponent": "-2",
      "subexpr": "\\frac{1}{\\sigma^{2}}",
      "chartScript": {
        "script": "pow(sigma, -2)",
        "variables": [
          "sigma"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "x",
      "to": "__f_2"
    },
    {
      "from": "__f_2",
      "to": "__equals_1"
    },
    {
      "from": "sigma",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__num_8",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "pi",
      "to": "__multiply_7",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_7",
      "to": "__power_6"
    },
    {
      "from": "__power_6",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_5",
      "to": "__power_4"
    },
    {
      "from": "__power_4",
      "to": "__multiply_3"
    },
    {
      "from": "e",
      "to": "__power_9"
    },
    {
      "from": "x",
      "to": "__add_13"
    },
    {
      "from": "mu",
      "to": "__negation_14"
    },
    {
      "from": "__negation_14",
      "to": "__add_13"
    },
    {
      "from": "__add_13",
      "to": "__power_12"
    },
    {
      "from": "__power_12",
      "to": "__multiply_11",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__num_16",
      "to": "__multiply_15",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "sigma",
      "to": "__power_17"
    },
    {
      "from": "__power_17",
      "to": "__multiply_15"
    },
    {
      "from": "__multiply_15",
      "to": "__multiply_11",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_11",
      "to": "__negation_10"
    },
    {
      "from": "__negation_10",
      "to": "__power_9",
      "role": "exp"
    },
    {
      "from": "__power_9",
      "to": "__multiply_3"
    },
    {
      "from": "__multiply_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __f_2{{"$f(\cdot)$"}}:::function
  x["$x$"]:::scalar
  __multiply_3(("$\times$")):::operator
  __power_4(("$\dfrac{1}{(\cdot)}$")):::operator
  __multiply_5(("$\times$")):::operator
  sigma["$\sigma$"]:::scalar
  __power_6(("${(\cdot)}^{1/2}$")):::operator
  __multiply_7(("$\times$")):::operator
  __num_8["$2$"]:::number
  pi["$\pi$"]:::constant
  __power_9(("$(\cdot)^{\cdot}$")):::operator
  e["$e$"]:::scalar
  __negation_10@{ shape: "flip-tri", label: "$-$" }
  __multiply_11(("$\times$")):::operator
  __power_12(("${(\cdot)}^{2}$")):::operator
  __add_13(("$+$")):::operator
  __negation_14@{ shape: "flip-tri", label: "$-$" }
  mu["$\mu$"]:::scalar
  __multiply_15(("$\times$")):::operator
  __num_16["$\frac{1}{2}$"]:::number
  __power_17(("${(\cdot)}^{-2}$")):::operator
  class __negation_10 operator
  class __negation_14 operator
  x --> __f_2
  __f_2 --> __equals_1
  sigma --> __multiply_5
  __num_8 --> __multiply_7
  pi --> __multiply_7
  __multiply_7 --> __power_6
  __power_6 --> __multiply_5
  __multiply_5 --> __power_4
  __power_4 -.-> __multiply_3
  e --> __power_9
  x --> __add_13
  mu --> __negation_14
  __negation_14 --> __add_13
  __add_13 --> __power_12
  __power_12 --> __multiply_11
  __num_16 --> __multiply_15
  sigma --> __power_17
  __power_17 -.-> __multiply_15
  __multiply_15 --> __multiply_11
  __multiply_11 --> __negation_10
  __negation_10 -->|exp| __power_9
  __power_9 --> __multiply_3
  __multiply_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#ef5350,stroke-width:3px
  linkStyle 3 stroke:#ef5350,stroke-width:3px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#42a5f5,stroke-width:1px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#aaa,stroke-width:2px
  linkStyle 14 stroke:#ef5350,stroke-width:3px
  linkStyle 15 stroke:#ef5350,stroke-width:3px
  linkStyle 16 stroke:#aaa,stroke-width:2px
  linkStyle 17 stroke:#42a5f5,stroke-width:2px
  linkStyle 18 stroke:#ef5350,stroke-width:3px
  linkStyle 19 stroke:#aaa,stroke-width:2px
  linkStyle 20 stroke:#aaa,stroke-width:2px
  linkStyle 21 stroke:#ef5350,stroke-width:3px
  linkStyle 22 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "P",
      "type": "scalar",
      "latex": "P",
      "subexpr": "P(X",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k)",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "s2___multiply_1",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\binom{n}{k} p^k (1-p)^{n-k}",
      "chartScript": {
        "script": "pow(p, k)*pow(1 - p, -k + n)*binomial(n, k)",
        "variables": [
          "k",
          "n",
          "p"
        ]
      }
    },
    {
      "id": "s2___choose_2",
      "type": "function",
      "op": "choose",
      "subexpr": "{\\binom{n}{k}}",
      "chartScript": {
        "script": "binomial(n, k)",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "n",
      "type": "scalar",
      "latex": "n",
      "subexpr": "n",
      "chartScript": {
        "script": "n",
        "variables": [
          "n"
        ]
      }
    },
    {
      "id": "s2___multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "p^{k} \\left(1 - p\\right)^{- k + n}",
      "chartScript": {
        "script": "pow(p, k)*pow(1 - p, -k + n)",
        "variables": [
          "k",
          "n",
          "p"
        ]
      }
    },
    {
      "id": "s2___power_4",
      "type": "operator",
      "op": "power",
      "subexpr": "p^{k}",
      "chartScript": {
        "script": "pow(p, k)",
        "variables": [
          "k",
          "p"
        ]
      }
    },
    {
      "id": "p",
      "type": "scalar",
      "latex": "p",
      "subexpr": "p",
      "chartScript": {
        "script": "p",
        "variables": [
          "p"
        ]
      }
    },
    {
      "id": "s2___power_5",
      "type": "operator",
      "op": "power",
      "subexpr": "\\left(1 - p\\right)^{- k + n}",
      "chartScript": {
        "script": "pow(1 - p, -k + n)",
        "variables": [
          "k",
          "n",
          "p"
        ]
      }
    },
    {
      "id": "s2___add_6",
      "type": "operator",
      "op": "add",
      "subexpr": "1 - p",
      "chartScript": {
        "script": "1 - p",
        "variables": [
          "p"
        ]
      }
    },
    {
      "id": "s2___num_7",
      "type": "number",
      "label": "1",
      "subexpr": "1",
      "chartScript": {
        "script": "1",
        "variables": []
      }
    },
    {
      "id": "s2___negation_8",
      "type": "operator",
      "op": "negation",
      "subexpr": "-p",
      "chartScript": {
        "script": "-p",
        "variables": [
          "p"
        ]
      }
    },
    {
      "id": "s2___add_9",
      "type": "operator",
      "op": "add",
      "subexpr": "n - k",
      "chartScript": {
        "script": "-k + n",
        "variables": [
          "k",
          "n"
        ]
      }
    },
    {
      "id": "s2___negation_10",
      "type": "operator",
      "op": "negation",
      "subexpr": "-k",
      "chartScript": {
        "script": "-k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__equals_1",
      "type": "operator",
      "op": "equals",
      "subexpr": "P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "n",
      "to": "s2___choose_2"
    },
    {
      "from": "k",
      "to": "s2___choose_2"
    },
    {
      "from": "s2___choose_2",
      "to": "s2___multiply_1"
    },
    {
      "from": "p",
      "to": "s2___power_4"
    },
    {
      "from": "k",
      "to": "s2___power_4"
    },
    {
      "from": "s2___power_4",
      "to": "s2___multiply_3"
    },
    {
      "from": "s2___num_7",
      "to": "s2___add_6"
    },
    {
      "from": "p",
      "to": "s2___negation_8"
    },
    {
      "from": "s2___negation_8",
      "to": "s2___add_6"
    },
    {
      "from": "s2___add_6",
      "to": "s2___power_5"
    },
    {
      "from": "n",
      "to": "s2___add_9"
    },
    {
      "from": "k",
      "to": "s2___negation_10"
    },
    {
      "from": "s2___negation_10",
      "to": "s2___add_9"
    },
    {
      "from": "s2___add_9",
      "to": "s2___power_5"
    },
    {
      "from": "s2___power_5",
      "to": "s2___multiply_3"
    },
    {
      "from": "s2___multiply_3",
      "to": "s2___multiply_1"
    },
    {
      "from": "P",
      "to": "__equals_1"
    },
    {
      "from": "k",
      "to": "__equals_1"
    },
    {
      "from": "s2___multiply_1",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  P["$P$"]:::scalar
  k["$k$"]:::scalar
  s2___multiply_1(("$\times$")):::operator
  s2___choose_2{{"$\binom{\cdot}{\cdot}$"}}:::function
  n["$n$"]:::scalar
  s2___multiply_3(("$\times$")):::operator
  s2___power_4(("$(\cdot)^{\cdot}$")):::operator
  p["$p$"]:::scalar
  s2___power_5(("$(\cdot)^{\cdot}$")):::operator
  s2___add_6(("$+$")):::operator
  s2___num_7["$1$"]:::number
  s2___negation_8@{ shape: "flip-tri", label: "$-$" }
  s2___add_9(("$+$")):::operator
  s2___negation_10@{ shape: "flip-tri", label: "$-$" }
  __equals_1(("$=$")):::operator
  class s2___negation_8 operator
  class s2___negation_10 operator
  n --> s2___choose_2
  k --> s2___choose_2
  s2___choose_2 --> s2___multiply_1
  p --> s2___power_4
  k --> s2___power_4
  s2___power_4 --> s2___multiply_3
  s2___num_7 --> s2___add_6
  p --> s2___negation_8
  s2___negation_8 --> s2___add_6
  s2___add_6 --> s2___power_5
  n --> s2___add_9
  k --> s2___negation_10
  s2___negation_10 --> s2___add_9
  s2___add_9 --> s2___power_5
  s2___power_5 --> s2___multiply_3
  s2___multiply_3 --> s2___multiply_1
  P --> __equals_1
  k --> __equals_1
  s2___multiply_1 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#ef5350,stroke-width:3px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#aaa,stroke-width:2px
  linkStyle 13 stroke:#aaa,stroke-width:2px
  linkStyle 14 stroke:#ef5350,stroke-width:3px
  linkStyle 15 stroke:#ef5350,stroke-width:3px
  linkStyle 16 stroke:#aaa,stroke-width:2px
  linkStyle 17 stroke:#aaa,stroke-width:2px
  linkStyle 18 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "P",
      "type": "scalar",
      "latex": "P",
      "subexpr": "P(X",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k)",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "s2___multiply_1",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{\\lambda^k e^{-\\lambda}}{k!}",
      "chartScript": {
        "script": "pow(lambda, k)*exp(-lambda)/factorial(k)",
        "variables": [
          "k",
          "lambda"
        ]
      }
    },
    {
      "id": "s2___multiply_2",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\lambda^{k} e^{- \\lambda}",
      "chartScript": {
        "script": "pow(lambda, k)*exp(-lambda)",
        "variables": [
          "k",
          "lambda"
        ]
      }
    },
    {
      "id": "s2___power_3",
      "type": "operator",
      "op": "power",
      "subexpr": "\\lambda^{k}",
      "chartScript": {
        "script": "pow(lambda, k)",
        "variables": [
          "k",
          "lambda"
        ]
      }
    },
    {
      "id": "lambda",
      "type": "scalar",
      "latex": "\\lambda",
      "subexpr": "\\lambda",
      "chartScript": {
        "script": "lambda",
        "variables": [
          "lambda"
        ]
      }
    },
    {
      "id": "s2___power_4",
      "type": "operator",
      "op": "power",
      "exponent": "-\\lambda",
      "subexpr": "e^{- \\lambda}",
      "chartScript": {
        "script": "exp(-lambda)",
        "variables": [
          "lambda"
        ]
      }
    },
    {
      "id": "e",
      "type": "scalar",
      "latex": "e",
      "subexpr": "e",
      "chartScript": {
        "script": "e",
        "variables": []
      }
    },
    {
      "id": "s2___power_5",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{k!}",
      "chartScript": {
        "script": "1/factorial(k)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "s2___factorial_6",
      "type": "operator",
      "op": "factorial",
      "subexpr": "k!",
      "chartScript": {
        "script": "factorial(k)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__equals_1",
      "type": "operator",
      "op": "equals",
      "subexpr": "P(X = k) = \\frac{\\lambda^k e^{-\\lambda}}{k!}",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "lambda",
      "to": "s2___power_3"
    },
    {
      "from": "k",
      "to": "s2___power_3"
    },
    {
      "from": "s2___power_3",
      "to": "s2___multiply_2"
    },
    {
      "from": "e",
      "to": "s2___power_4"
    },
    {
      "from": "s2___power_4",
      "to": "s2___multiply_2"
    },
    {
      "from": "s2___multiply_2",
      "to": "s2___multiply_1"
    },
    {
      "from": "k",
      "to": "s2___factorial_6"
    },
    {
      "from": "s2___factorial_6",
      "to": "s2___power_5"
    },
    {
      "from": "s2___power_5",
      "to": "s2___multiply_1"
    },
    {
      "from": "P",
      "to": "__equals_1"
    },
    {
      "from": "k",
      "to": "__equals_1"
    },
    {
      "from": "s2___multiply_1",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  P["$P$"]:::scalar
  k["$k$"]:::scalar
  s2___multiply_1(("$\times$")):::operator
  s2___multiply_2(("$\times$")):::operator
  s2___power_3(("$(\cdot)^{\cdot}$")):::operator
  lambda["$\lambda$"]:::scalar
  s2___power_4(("${(\cdot)}^{-\lambda}$")):::operator
  e["$e$"]:::scalar
  s2___power_5(("$\dfrac{1}{(\cdot)}$")):::operator
  s2___factorial_6(("$(\cdot)!$")):::operator
  __equals_1(("$=$")):::operator
  lambda --> s2___power_3
  k --> s2___power_3
  s2___power_3 --> s2___multiply_2
  e --> s2___power_4
  s2___power_4 -.-> s2___multiply_2
  s2___multiply_2 --> s2___multiply_1
  k --> s2___factorial_6
  s2___factorial_6 --> s2___power_5
  s2___power_5 -.-> s2___multiply_1
  P --> __equals_1
  k --> __equals_1
  s2___multiply_1 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#ef5350,stroke-width:3px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#42a5f5,stroke-width:1px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#42a5f5,stroke-width:1px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "P(A \\cap B)  =  P(A) P(B \\mid A)",
      "chartScript": {
        "script": "-P(A)*P(A*B*mid) + P(A*B*cap)",
        "variables": [
          "A",
          "B",
          "cap",
          "mid"
        ]
      }
    },
    {
      "id": "__P_2",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\cap B \\right)}",
      "chartScript": {
        "script": "A*B*P*cap",
        "variables": [
          "A",
          "B",
          "P",
          "cap"
        ]
      }
    },
    {
      "id": "A",
      "type": "scalar",
      "latex": "A",
      "subexpr": "A",
      "chartScript": {
        "script": "A",
        "variables": [
          "A"
        ]
      }
    },
    {
      "id": "B",
      "type": "scalar",
      "latex": "B",
      "subexpr": "B",
      "chartScript": {
        "script": "B",
        "variables": [
          "B"
        ]
      }
    },
    {
      "id": "__intersection_3",
      "type": "operator",
      "emoji": "∩",
      "op": "intersection",
      "subexpr": "A \\cap B",
      "chartScript": {
        "script": "A*B*cap",
        "variables": [
          "A",
          "B",
          "cap"
        ]
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "P{\\left(A \\right)} P{\\left(B \\mid A \\right)}",
      "chartScript": {
        "script": "pow(A, 2)*B*pow(P, 2)*mid",
        "variables": [
          "A",
          "B",
          "P",
          "mid"
        ]
      }
    },
    {
      "id": "__P_5",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\right)}",
      "chartScript": {
        "script": "A*P",
        "variables": [
          "A",
          "P"
        ]
      }
    },
    {
      "id": "__P_6",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(B \\mid A \\right)}",
      "chartScript": {
        "script": "A*B*P*mid",
        "variables": [
          "A",
          "B",
          "P",
          "mid"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "A",
      "to": "__intersection_3"
    },
    {
      "from": "B",
      "to": "__intersection_3"
    },
    {
      "from": "__intersection_3",
      "to": "__P_2"
    },
    {
      "from": "__P_2",
      "to": "__equals_1"
    },
    {
      "from": "A",
      "to": "__P_5"
    },
    {
      "from": "__P_5",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "B",
      "to": "__P_6"
    },
    {
      "from": "A",
      "to": "__P_6",
      "role": "condition"
    },
    {
      "from": "__P_6",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_4",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __P_2{{"$P(\cdot)$"}}:::function
  A["$A$"]:::scalar
  B["$B$"]:::scalar
  __intersection_3(("$\cap$")):::operator
  __multiply_4(("$\times$")):::operator
  __P_5{{"$P(\cdot)$"}}:::function
  __P_6{{"$P(\cdot\mid \cdot)$"}}:::function
  A --> __intersection_3
  B --> __intersection_3
  __intersection_3 --> __P_2
  __P_2 --> __equals_1
  A --> __P_5
  __P_5 --> __multiply_4
  B --> __P_6
  A -->|condition| __P_6
  __P_6 --> __multiply_4
  __multiply_4 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#ef5350,stroke-width:3px
  linkStyle 9 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "E(aX + b) = aE(X) + b",
      "chartScript": {
        "script": "-a*E(X) - b + E(X*a + b)",
        "variables": [
          "X",
          "a",
          "b"
        ]
      }
    },
    {
      "id": "__E_2",
      "type": "function",
      "op": "E",
      "subexpr": "E{\\left(X a + b \\right)}",
      "chartScript": {
        "script": "E*(X*a + b)",
        "variables": [
          "E",
          "X",
          "a",
          "b"
        ]
      }
    },
    {
      "id": "__add_3",
      "type": "operator",
      "op": "add",
      "subexpr": "a X + b",
      "chartScript": {
        "script": "X*a + b",
        "variables": [
          "X",
          "a",
          "b"
        ]
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "a X",
      "chartScript": {
        "script": "X*a",
        "variables": [
          "X",
          "a"
        ]
      }
    },
    {
      "id": "a",
      "type": "scalar",
      "latex": "a",
      "subexpr": "a",
      "chartScript": {
        "script": "a",
        "variables": [
          "a"
        ]
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "b",
      "type": "scalar",
      "latex": "b",
      "subexpr": "b",
      "chartScript": {
        "script": "b",
        "variables": [
          "b"
        ]
      }
    },
    {
      "id": "__add_5",
      "type": "operator",
      "op": "add",
      "subexpr": "a E{\\left(X \\right)} + b",
      "chartScript": {
        "script": "E*X*a + b",
        "variables": [
          "E",
          "X",
          "a",
          "b"
        ]
      }
    },
    {
      "id": "__multiply_6",
      "type": "operator",
      "op": "multiply",
      "subexpr": "a E{\\left(X \\right)}",
      "chartScript": {
        "script": "E*X*a",
        "variables": [
          "E",
          "X",
          "a"
        ]
      }
    },
    {
      "id": "__E_7",
      "type": "function",
      "op": "E",
      "subexpr": "E{\\left(X \\right)}",
      "chartScript": {
        "script": "E*X",
        "variables": [
          "E",
          "X"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "a",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "X",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_4",
      "to": "__add_3"
    },
    {
      "from": "b",
      "to": "__add_3"
    },
    {
      "from": "__add_3",
      "to": "__E_2"
    },
    {
      "from": "__E_2",
      "to": "__equals_1"
    },
    {
      "from": "a",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "X",
      "to": "__E_7"
    },
    {
      "from": "__E_7",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_6",
      "to": "__add_5"
    },
    {
      "from": "b",
      "to": "__add_5"
    },
    {
      "from": "__add_5",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __E_2{{"$E(\cdot)$"}}:::function
  __add_3(("$+$")):::operator
  __multiply_4(("$\times$")):::operator
  a["$a$"]:::scalar
  X["$X$"]:::scalar
  b["$b$"]:::scalar
  __add_5(("$+$")):::operator
  __multiply_6(("$\times$")):::operator
  __E_7{{"$E(\cdot)$"}}:::function
  a --> __multiply_4
  X --> __multiply_4
  __multiply_4 --> __add_3
  b --> __add_3
  __add_3 --> __E_2
  __E_2 --> __equals_1
  a --> __multiply_6
  X --> __E_7
  __E_7 --> __multiply_6
  __multiply_6 --> __add_5
  b --> __add_5
  __add_5 --> __equals_1
  linkStyle 0 stroke:#ef5350,stroke-width:3px
  linkStyle 1 stroke:#ef5350,stroke-width:3px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#ef5350,stroke-width:3px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "\\sigma_{XY} = E(XY) - \\mu_X \\mu_Y",
      "chartScript": {
        "script": "mu_X*mu_Y + sigma_X*Y - E(X*Y)",
        "variables": [
          "X",
          "Y",
          "mu_X",
          "mu_Y",
          "sigma_X*Y"
        ]
      }
    },
    {
      "id": "sigma_XY",
      "type": "scalar",
      "latex": "\\sigma_{XY}",
      "subexpr": "\\sigma_{XY}",
      "chartScript": {
        "script": "sigma_X*Y",
        "variables": [
          "sigma_X*Y"
        ]
      }
    },
    {
      "id": "__add_2",
      "type": "operator",
      "op": "add",
      "subexpr": "-\\mu_{X} \\mu_{Y} + E{\\left(X Y \\right)}",
      "chartScript": {
        "script": "E*X*Y - mu_X*mu_Y",
        "variables": [
          "E",
          "X",
          "Y",
          "mu_X",
          "mu_Y"
        ]
      }
    },
    {
      "id": "__E_3",
      "type": "function",
      "op": "E",
      "subexpr": "E{\\left(X Y \\right)}",
      "chartScript": {
        "script": "E*X*Y",
        "variables": [
          "E",
          "X",
          "Y"
        ]
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "X Y",
      "chartScript": {
        "script": "X*Y",
        "variables": [
          "X",
          "Y"
        ]
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "Y",
      "type": "scalar",
      "latex": "Y",
      "subexpr": "Y",
      "chartScript": {
        "script": "Y",
        "variables": [
          "Y"
        ]
      }
    },
    {
      "id": "__negation_5",
      "type": "operator",
      "op": "negation",
      "subexpr": "-\\mu_{X} \\mu_{Y}",
      "chartScript": {
        "script": "-mu_X*mu_Y",
        "variables": [
          "mu_X",
          "mu_Y"
        ]
      }
    },
    {
      "id": "__multiply_6",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\mu_{X} \\mu_{Y}",
      "chartScript": {
        "script": "mu_X*mu_Y",
        "variables": [
          "mu_X",
          "mu_Y"
        ]
      }
    },
    {
      "id": "mu_X",
      "type": "scalar",
      "latex": "\\mu_{X}",
      "subexpr": "\\mu_{X}",
      "chartScript": {
        "script": "mu_X",
        "variables": [
          "mu_X"
        ]
      }
    },
    {
      "id": "mu_Y",
      "type": "scalar",
      "latex": "\\mu_{Y}",
      "subexpr": "\\mu_{Y}",
      "chartScript": {
        "script": "mu_Y",
        "variables": [
          "mu_Y"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "sigma_XY",
      "to": "__equals_1"
    },
    {
      "from": "X",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "Y",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_4",
      "to": "__E_3"
    },
    {
      "from": "__E_3",
      "to": "__add_2"
    },
    {
      "from": "mu_X",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "mu_Y",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_6",
      "to": "__negation_5"
    },
    {
      "from": "__negation_5",
      "to": "__add_2"
    },
    {
      "from": "__add_2",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  sigma_XY["$\sigma_{XY}$"]:::scalar
  __add_2(("$+$")):::operator
  __E_3{{"$E(\cdot)$"}}:::function
  __multiply_4(("$\times$")):::operator
  X["$X$"]:::scalar
  Y["$Y$"]:::scalar
  __negation_5@{ shape: "flip-tri", label: "$-$" }
  __multiply_6(("$\times$")):::operator
  mu_X["$\mu_{X}$"]:::scalar
  mu_Y["$\mu_{Y}$"]:::scalar
  class __negation_5 operator
  sigma_XY --> __equals_1
  X --> __multiply_4
  Y --> __multiply_4
  __multiply_4 --> __E_3
  __E_3 --> __add_2
  mu_X --> __multiply_6
  mu_Y --> __multiply_6
  __multiply_6 --> __negation_5
  __negation_5 --> __add_2
  __add_2 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#ef5350,stroke-width:3px
  linkStyle 2 stroke:#ef5350,stroke-width:3px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__P_1",
      "type": "function",
      "op": "P",
      "subexpr": "P(X \\geq a)",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "a",
      "type": "scalar",
      "latex": "a",
      "subexpr": "a",
      "chartScript": {
        "script": "a",
        "variables": [
          "a"
        ]
      }
    },
    {
      "id": "__greater_equal_2",
      "type": "relation",
      "emoji": "≥",
      "op": "greater_equal",
      "subexpr": "X \\geq a",
      "chartScript": {
        "script": "X - a",
        "variables": [
          "X",
          "a"
        ]
      }
    },
    {
      "id": "__multiply_3",
      "type": "operator",
      "op": "multiply",
      "subexpr": "\\frac{E(X)}{a}",
      "chartScript": {
        "script": "E(X)/a",
        "variables": [
          "X",
          "a"
        ]
      }
    },
    {
      "id": "__E_4",
      "type": "function",
      "op": "E",
      "subexpr": "E{\\left(X \\right)}",
      "chartScript": {
        "script": "E*X",
        "variables": [
          "E",
          "X"
        ]
      }
    },
    {
      "id": "__power_5",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{a}",
      "chartScript": {
        "script": "1/a",
        "variables": [
          "a"
        ]
      }
    },
    {
      "id": "__less_equal_6",
      "type": "relation",
      "label": "less than or equal to",
      "emoji": "≤",
      "op": "less_equal",
      "subexpr": "P(X \\geq a) \\leq \\frac{E(X)}{a}",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "X",
      "to": "__greater_equal_2",
      "role": "lhs"
    },
    {
      "from": "a",
      "to": "__greater_equal_2",
      "role": "rhs"
    },
    {
      "from": "__greater_equal_2",
      "to": "__P_1",
      "role": "assertion"
    },
    {
      "from": "X",
      "to": "__E_4"
    },
    {
      "from": "__E_4",
      "to": "__multiply_3",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "a",
      "to": "__power_5"
    },
    {
      "from": "__power_5",
      "to": "__multiply_3"
    },
    {
      "from": "__P_1",
      "to": "__less_equal_6",
      "role": "lhs"
    },
    {
      "from": "__multiply_3",
      "to": "__less_equal_6",
      "role": "rhs"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __P_1{{"$P(\ldots)$"}}:::function
  X["$X$"]:::scalar
  a["$a$"]:::scalar
  __greater_equal_2{"≥"}:::relation
  __multiply_3(("$\times$")):::operator
  __E_4{{"$E(\cdot)$"}}:::function
  __power_5(("$\dfrac{1}{(\cdot)}$")):::operator
  __less_equal_6{"≤"}:::relation
  X -->|lhs| __greater_equal_2
  a -->|rhs| __greater_equal_2
  __greater_equal_2 -->|assertion| __P_1
  X --> __E_4
  __E_4 --> __multiply_3
  a --> __power_5
  __power_5 -.-> __multiply_3
  __P_1 -->|lhs| __less_equal_6
  __multiply_3 -->|rhs| __less_equal_6
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#42a5f5,stroke-width:1px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "P(A^c) = 1 - P(A)",
      "chartScript": {
        "script": "P(A) + P(A**c) - 1",
        "variables": [
          "A",
          "c"
        ]
      }
    },
    {
      "id": "__P_2",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A^{c} \\right)}",
      "chartScript": {
        "script": "pow(A, c)*P",
        "variables": [
          "A",
          "P",
          "c"
        ]
      }
    },
    {
      "id": "__power_3",
      "type": "operator",
      "op": "power",
      "subexpr": "A^{c}",
      "chartScript": {
        "script": "pow(A, c)",
        "variables": [
          "A",
          "c"
        ]
      }
    },
    {
      "id": "A",
      "type": "scalar",
      "latex": "A",
      "subexpr": "A",
      "chartScript": {
        "script": "A",
        "variables": [
          "A"
        ]
      }
    },
    {
      "id": "c",
      "type": "scalar",
      "latex": "c",
      "subexpr": "c",
      "chartScript": {
        "script": "c",
        "variables": [
          "c"
        ]
      }
    },
    {
      "id": "__add_4",
      "type": "operator",
      "op": "add",
      "subexpr": "1 - P{\\left(A \\right)}",
      "chartScript": {
        "script": "-A*P + 1",
        "variables": [
          "A",
          "P"
        ]
      }
    },
    {
      "id": "__num_5",
      "type": "number",
      "label": "1",
      "subexpr": "1",
      "chartScript": {
        "script": "1",
        "variables": []
      }
    },
    {
      "id": "__negation_6",
      "type": "operator",
      "op": "negation",
      "subexpr": "-P{\\left(A \\right)}",
      "chartScript": {
        "script": "-A*P",
        "variables": [
          "A",
          "P"
        ]
      }
    },
    {
      "id": "__P_7",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\right)}",
      "chartScript": {
        "script": "A*P",
        "variables": [
          "A",
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "A",
      "to": "__power_3"
    },
    {
      "from": "c",
      "to": "__power_3",
      "role": "exp"
    },
    {
      "from": "__power_3",
      "to": "__P_2"
    },
    {
      "from": "__P_2",
      "to": "__equals_1"
    },
    {
      "from": "__num_5",
      "to": "__add_4"
    },
    {
      "from": "A",
      "to": "__P_7"
    },
    {
      "from": "__P_7",
      "to": "__negation_6"
    },
    {
      "from": "__negation_6",
      "to": "__add_4"
    },
    {
      "from": "__add_4",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __P_2{{"$P(\cdot)$"}}:::function
  __power_3(("$(\cdot)^{\cdot}$")):::operator
  A["$A$"]:::scalar
  c["$c$"]:::scalar
  __add_4(("$+$")):::operator
  __num_5["$1$"]:::number
  __negation_6@{ shape: "flip-tri", label: "$-$" }
  __P_7{{"$P(\cdot)$"}}:::function
  class __negation_6 operator
  A --> __power_3
  c -->|exp| __power_3
  __power_3 --> __P_2
  __P_2 --> __equals_1
  __num_5 --> __add_4
  A --> __P_7
  __P_7 --> __negation_6
  __negation_6 --> __add_4
  __add_4 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__P_1",
      "type": "function",
      "op": "P",
      "subexpr": "P(|X - \\mu| \\geq k\\sigma)",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "__abs_3",
      "type": "function",
      "op": "abs",
      "subexpr": "\\left|{X - \\mu}\\right|"
    },
    {
      "id": "__add_4",
      "type": "operator",
      "op": "add",
      "subexpr": "X - \\mu",
      "chartScript": {
        "script": "X - mu",
        "variables": [
          "X",
          "mu"
        ]
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "__negation_5",
      "type": "operator",
      "op": "negation",
      "subexpr": "-\\mu",
      "chartScript": {
        "script": "-mu",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "mu",
      "type": "scalar",
      "latex": "\\mu",
      "subexpr": "\\mu",
      "chartScript": {
        "script": "mu",
        "variables": [
          "mu"
        ]
      }
    },
    {
      "id": "__multiply_6",
      "type": "operator",
      "op": "multiply",
      "subexpr": "k \\sigma",
      "chartScript": {
        "script": "k*sigma",
        "variables": [
          "k",
          "sigma"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "sigma",
      "type": "scalar",
      "latex": "\\sigma",
      "subexpr": "\\sigma",
      "chartScript": {
        "script": "sigma",
        "variables": [
          "sigma"
        ]
      }
    },
    {
      "id": "__greater_equal_2",
      "type": "relation",
      "emoji": "≥",
      "op": "greater_equal",
      "subexpr": "\\left|{X - \\mu}\\right| \\geq k \\sigma"
    },
    {
      "id": "__power_7",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{k^2}",
      "chartScript": {
        "script": "pow(k, -2)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__power_8",
      "type": "operator",
      "op": "power",
      "exponent": "2",
      "subexpr": "k^{2}",
      "chartScript": {
        "script": "pow(k, 2)",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "__less_equal_9",
      "type": "relation",
      "label": "less than or equal to",
      "emoji": "≤",
      "op": "less_equal",
      "subexpr": "P(|X - \\mu| \\geq k\\sigma) \\leq \\frac{1}{k^2}",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "X",
      "to": "__add_4"
    },
    {
      "from": "mu",
      "to": "__negation_5"
    },
    {
      "from": "__negation_5",
      "to": "__add_4"
    },
    {
      "from": "__add_4",
      "to": "__abs_3"
    },
    {
      "from": "__abs_3",
      "to": "__greater_equal_2",
      "role": "lhs"
    },
    {
      "from": "k",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "sigma",
      "to": "__multiply_6",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_6",
      "to": "__greater_equal_2",
      "role": "rhs"
    },
    {
      "from": "__greater_equal_2",
      "to": "__P_1",
      "role": "assertion"
    },
    {
      "from": "k",
      "to": "__power_8"
    },
    {
      "from": "__power_8",
      "to": "__power_7"
    },
    {
      "from": "__P_1",
      "to": "__less_equal_9",
      "role": "lhs"
    },
    {
      "from": "__power_7",
      "to": "__less_equal_9",
      "role": "rhs"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __P_1{{"$P(\ldots)$"}}:::function
  __abs_3{{"$|\cdot|$"}}:::function
  __add_4(("$+$")):::operator
  X["$X$"]:::scalar
  __negation_5@{ shape: "flip-tri", label: "$-$" }
  mu["$\mu$"]:::scalar
  __multiply_6(("$\times$")):::operator
  k["$k$"]:::scalar
  sigma["$\sigma$"]:::scalar
  __greater_equal_2{"≥"}:::relation
  __power_7(("$\dfrac{1}{(\cdot)}$")):::operator
  __power_8(("${(\cdot)}^{2}$")):::operator
  __less_equal_9{"≤"}:::relation
  class __negation_5 operator
  X --> __add_4
  mu --> __negation_5
  __negation_5 --> __add_4
  __add_4 --> __abs_3
  __abs_3 -->|lhs| __greater_equal_2
  k --> __multiply_6
  sigma --> __multiply_6
  __multiply_6 -->|rhs| __greater_equal_2
  __greater_equal_2 -->|assertion| __P_1
  k --> __power_8
  __power_8 --> __power_7
  __P_1 -->|lhs| __less_equal_9
  __power_7 -.->|rhs| __less_equal_9
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#ef5350,stroke-width:3px
  linkStyle 7 stroke:#aaa,stroke-width:2px
  linkStyle 8 stroke:#aaa,stroke-width:2px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#ef5350,stroke-width:6px
  linkStyle 11 stroke:#aaa,stroke-width:2px
  linkStyle 12 stroke:#42a5f5,stroke-width:1px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "P(1 < X \\leq 10) = 0.5",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "__P_2",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(1 < X \\leq 10 \\right)}",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "__num_3",
      "type": "number",
      "label": "1",
      "subexpr": "1",
      "chartScript": {
        "script": "1",
        "variables": []
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "__num_4",
      "type": "number",
      "label": "10",
      "subexpr": "10",
      "chartScript": {
        "script": "10",
        "variables": []
      }
    },
    {
      "id": "__less_than_5",
      "type": "relation",
      "emoji": "<",
      "op": "less_than",
      "subexpr": "1 < X",
      "chartScript": {
        "script": "1 - X",
        "variables": [
          "X"
        ]
      }
    },
    {
      "id": "__less_equal_6",
      "type": "relation",
      "emoji": "≤",
      "op": "less_equal",
      "subexpr": "1 < X \\leq 10"
    },
    {
      "id": "__num_7",
      "type": "number",
      "label": "0.5",
      "subexpr": "0.5",
      "chartScript": {
        "script": "0.500000000000000",
        "variables": []
      }
    }
  ],
  "edges": [
    {
      "from": "__num_3",
      "to": "__less_than_5",
      "role": "lhs"
    },
    {
      "from": "X",
      "to": "__less_than_5",
      "role": "rhs"
    },
    {
      "from": "__less_than_5",
      "to": "__less_equal_6",
      "role": "lhs"
    },
    {
      "from": "__num_4",
      "to": "__less_equal_6",
      "role": "rhs"
    },
    {
      "from": "__less_equal_6",
      "to": "__P_2",
      "role": "assertion"
    },
    {
      "from": "__P_2",
      "to": "__equals_1"
    },
    {
      "from": "__num_7",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __P_2{{"$P(\ldots)$"}}:::function
  __num_3["$1$"]:::number
  X["$X$"]:::scalar
  __num_4["$10$"]:::number
  __less_than_5{"$<$"}:::relation
  __less_equal_6{"≤"}:::relation
  __num_7["$0.5$"]:::number
  __num_3 -->|lhs| __less_than_5
  X -->|rhs| __less_than_5
  __less_than_5 -->|lhs| __less_equal_6
  __num_4 -->|rhs| __less_equal_6
  __less_equal_6 -->|assertion| __P_2
  __P_2 --> __equals_1
  __num_7 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "f(x) = \\frac{1}{b - a}",
      "chartScript": {
        "script": "f(x) - 1/(-a + b)",
        "variables": [
          "a",
          "b",
          "x"
        ]
      }
    },
    {
      "id": "__f_2",
      "type": "function",
      "op": "f",
      "subexpr": "f{\\left(x \\right)}",
      "chartScript": {
        "script": "f*x",
        "variables": [
          "f",
          "x"
        ]
      }
    },
    {
      "id": "x",
      "type": "scalar",
      "latex": "x",
      "subexpr": "x",
      "chartScript": {
        "script": "x",
        "variables": [
          "x"
        ]
      }
    },
    {
      "id": "__power_3",
      "type": "operator",
      "latex": "\\dfrac{1}{(\\cdot)}",
      "op": "power",
      "exponent": "-1",
      "subexpr": "\\frac{1}{- a + b}",
      "chartScript": {
        "script": "1/(-a + b)",
        "variables": [
          "a",
          "b"
        ]
      }
    },
    {
      "id": "__add_4",
      "type": "operator",
      "op": "add",
      "subexpr": "b - a",
      "chartScript": {
        "script": "-a + b",
        "variables": [
          "a",
          "b"
        ]
      }
    },
    {
      "id": "b",
      "type": "scalar",
      "latex": "b",
      "subexpr": "b",
      "chartScript": {
        "script": "b",
        "variables": [
          "b"
        ]
      }
    },
    {
      "id": "__negation_5",
      "type": "operator",
      "op": "negation",
      "subexpr": "-a",
      "chartScript": {
        "script": "-a",
        "variables": [
          "a"
        ]
      }
    },
    {
      "id": "a",
      "type": "scalar",
      "latex": "a",
      "subexpr": "a",
      "chartScript": {
        "script": "a",
        "variables": [
          "a"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "x",
      "to": "__f_2"
    },
    {
      "from": "__f_2",
      "to": "__equals_1"
    },
    {
      "from": "b",
      "to": "__add_4"
    },
    {
      "from": "a",
      "to": "__negation_5"
    },
    {
      "from": "__negation_5",
      "to": "__add_4"
    },
    {
      "from": "__add_4",
      "to": "__power_3"
    },
    {
      "from": "__power_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __f_2{{"$f(\cdot)$"}}:::function
  x["$x$"]:::scalar
  __power_3(("$\dfrac{1}{(\cdot)}$")):::operator
  __add_4(("$+$")):::operator
  b["$b$"]:::scalar
  __negation_5@{ shape: "flip-tri", label: "$-$" }
  a["$a$"]:::scalar
  class __negation_5 operator
  x --> __f_2
  __f_2 --> __equals_1
  b --> __add_4
  a --> __negation_5
  __negation_5 --> __add_4
  __add_4 --> __power_3
  __power_3 -.-> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#42a5f5,stroke-width:1px

{
  "nodes": [
    {
      "id": "P",
      "type": "scalar",
      "latex": "P",
      "subexpr": "P(X",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    },
    {
      "id": "k",
      "type": "scalar",
      "latex": "k",
      "subexpr": "k)",
      "chartScript": {
        "script": "k",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "s2___multiply_1",
      "type": "operator",
      "op": "multiply",
      "subexpr": "(1 - p)^{k-1} p",
      "chartScript": {
        "script": "p*pow(1 - p, k - 1)",
        "variables": [
          "k",
          "p"
        ]
      }
    },
    {
      "id": "s2___power_2",
      "type": "operator",
      "op": "power",
      "subexpr": "\\left(1 - p\\right)^{k - 1}",
      "chartScript": {
        "script": "pow(1 - p, k - 1)",
        "variables": [
          "k",
          "p"
        ]
      }
    },
    {
      "id": "s2___add_3",
      "type": "operator",
      "op": "add",
      "subexpr": "1 - p",
      "chartScript": {
        "script": "1 - p",
        "variables": [
          "p"
        ]
      }
    },
    {
      "id": "s2___num_4",
      "type": "number",
      "label": "1",
      "subexpr": "1",
      "chartScript": {
        "script": "1",
        "variables": []
      }
    },
    {
      "id": "s2___negation_5",
      "type": "operator",
      "op": "negation",
      "subexpr": "-p",
      "chartScript": {
        "script": "-p",
        "variables": [
          "p"
        ]
      }
    },
    {
      "id": "p",
      "type": "scalar",
      "latex": "p",
      "subexpr": "p",
      "chartScript": {
        "script": "p",
        "variables": [
          "p"
        ]
      }
    },
    {
      "id": "s2___add_6",
      "type": "operator",
      "op": "add",
      "subexpr": "k - 1",
      "chartScript": {
        "script": "k - 1",
        "variables": [
          "k"
        ]
      }
    },
    {
      "id": "s2___num_7",
      "type": "number",
      "label": "-1",
      "subexpr": "-1",
      "chartScript": {
        "script": "-1",
        "variables": []
      }
    },
    {
      "id": "__equals_1",
      "type": "operator",
      "op": "equals",
      "subexpr": "P(X = k) = (1 - p)^{k-1} p",
      "chartScript": {
        "script": "P",
        "variables": [
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "s2___num_4",
      "to": "s2___add_3"
    },
    {
      "from": "p",
      "to": "s2___negation_5"
    },
    {
      "from": "s2___negation_5",
      "to": "s2___add_3"
    },
    {
      "from": "s2___add_3",
      "to": "s2___power_2"
    },
    {
      "from": "k",
      "to": "s2___add_6"
    },
    {
      "from": "s2___num_7",
      "to": "s2___add_6"
    },
    {
      "from": "s2___add_6",
      "to": "s2___power_2"
    },
    {
      "from": "s2___power_2",
      "to": "s2___multiply_1"
    },
    {
      "from": "p",
      "to": "s2___multiply_1"
    },
    {
      "from": "P",
      "to": "__equals_1"
    },
    {
      "from": "k",
      "to": "__equals_1"
    },
    {
      "from": "s2___multiply_1",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  P["$P$"]:::scalar
  k["$k$"]:::scalar
  s2___multiply_1(("$\times$")):::operator
  s2___power_2(("$(\cdot)^{\cdot}$")):::operator
  s2___add_3(("$+$")):::operator
  s2___num_4["$1$"]:::number
  s2___negation_5@{ shape: "flip-tri", label: "$-$" }
  p["$p$"]:::scalar
  s2___add_6(("$+$")):::operator
  s2___num_7["$-1$"]:::number
  __equals_1(("$=$")):::operator
  class s2___negation_5 operator
  s2___num_4 --> s2___add_3
  p --> s2___negation_5
  s2___negation_5 --> s2___add_3
  s2___add_3 --> s2___power_2
  k --> s2___add_6
  s2___num_7 --> s2___add_6
  s2___add_6 --> s2___power_2
  s2___power_2 --> s2___multiply_1
  p --> s2___multiply_1
  P --> __equals_1
  k --> __equals_1
  s2___multiply_1 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#ef5350,stroke-width:3px
  linkStyle 9 stroke:#aaa,stroke-width:2px
  linkStyle 10 stroke:#aaa,stroke-width:2px
  linkStyle 11 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "P(A \\cap B)  =  P(A) \\cdot P(B)",
      "chartScript": {
        "script": "-P(A)*P(B) + P(A*B*cap)",
        "variables": [
          "A",
          "B",
          "cap"
        ]
      }
    },
    {
      "id": "__P_2",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\cap B \\right)}",
      "chartScript": {
        "script": "A*B*P*cap",
        "variables": [
          "A",
          "B",
          "P",
          "cap"
        ]
      }
    },
    {
      "id": "A",
      "type": "scalar",
      "latex": "A",
      "subexpr": "A",
      "chartScript": {
        "script": "A",
        "variables": [
          "A"
        ]
      }
    },
    {
      "id": "B",
      "type": "scalar",
      "latex": "B",
      "subexpr": "B",
      "chartScript": {
        "script": "B",
        "variables": [
          "B"
        ]
      }
    },
    {
      "id": "__intersection_3",
      "type": "operator",
      "emoji": "∩",
      "op": "intersection",
      "subexpr": "A \\cap B",
      "chartScript": {
        "script": "A*B*cap",
        "variables": [
          "A",
          "B",
          "cap"
        ]
      }
    },
    {
      "id": "__multiply_4",
      "type": "operator",
      "op": "multiply",
      "subexpr": "P{\\left(A \\right)} P{\\left(B \\right)}",
      "chartScript": {
        "script": "A*B*pow(P, 2)",
        "variables": [
          "A",
          "B",
          "P"
        ]
      }
    },
    {
      "id": "__P_5",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(A \\right)}",
      "chartScript": {
        "script": "A*P",
        "variables": [
          "A",
          "P"
        ]
      }
    },
    {
      "id": "__P_6",
      "type": "function",
      "op": "P",
      "subexpr": "P{\\left(B \\right)}",
      "chartScript": {
        "script": "B*P",
        "variables": [
          "B",
          "P"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "A",
      "to": "__intersection_3"
    },
    {
      "from": "B",
      "to": "__intersection_3"
    },
    {
      "from": "__intersection_3",
      "to": "__P_2"
    },
    {
      "from": "__P_2",
      "to": "__equals_1"
    },
    {
      "from": "A",
      "to": "__P_5"
    },
    {
      "from": "__P_5",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "B",
      "to": "__P_6"
    },
    {
      "from": "__P_6",
      "to": "__multiply_4",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_4",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __P_2{{"$P(\cdot)$"}}:::function
  A["$A$"]:::scalar
  B["$B$"]:::scalar
  __intersection_3(("$\cap$")):::operator
  __multiply_4(("$\times$")):::operator
  __P_5{{"$P(\cdot)$"}}:::function
  __P_6{{"$P(\cdot)$"}}:::function
  A --> __intersection_3
  B --> __intersection_3
  __intersection_3 --> __P_2
  __P_2 --> __equals_1
  A --> __P_5
  __P_5 --> __multiply_4
  B --> __P_6
  __P_6 --> __multiply_4
  __multiply_4 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#aaa,stroke-width:2px
  linkStyle 4 stroke:#aaa,stroke-width:2px
  linkStyle 5 stroke:#ef5350,stroke-width:3px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#ef5350,stroke-width:3px
  linkStyle 8 stroke:#aaa,stroke-width:2px

{
  "nodes": [
    {
      "id": "__equals_1",
      "type": "relation",
      "op": "equals",
      "subexpr": "M_X(t) = E(e^{tX})",
      "chartScript": {
        "script": "-E(exp(X*t)) + M_X(t)",
        "variables": [
          "X",
          "t"
        ]
      }
    },
    {
      "id": "__M_X_2",
      "type": "function",
      "op": "M_{X}",
      "subexpr": "M_{X}{\\left(t \\right)}",
      "chartScript": {
        "script": "M_X*t",
        "variables": [
          "M_X",
          "t"
        ]
      }
    },
    {
      "id": "t",
      "type": "scalar",
      "latex": "t",
      "subexpr": "t",
      "chartScript": {
        "script": "t",
        "variables": [
          "t"
        ]
      }
    },
    {
      "id": "__E_3",
      "type": "function",
      "op": "E",
      "subexpr": "E{\\left(e^{X t} \\right)}",
      "chartScript": {
        "script": "E*exp(X*t)",
        "variables": [
          "E",
          "X",
          "t"
        ]
      }
    },
    {
      "id": "__power_4",
      "type": "operator",
      "op": "power",
      "subexpr": "e^{X t}",
      "chartScript": {
        "script": "exp(X*t)",
        "variables": [
          "X",
          "t"
        ]
      }
    },
    {
      "id": "e",
      "type": "scalar",
      "latex": "e",
      "subexpr": "e",
      "chartScript": {
        "script": "e",
        "variables": []
      }
    },
    {
      "id": "__multiply_5",
      "type": "operator",
      "op": "multiply",
      "subexpr": "X t",
      "chartScript": {
        "script": "X*t",
        "variables": [
          "X",
          "t"
        ]
      }
    },
    {
      "id": "X",
      "type": "scalar",
      "latex": "X",
      "subexpr": "X",
      "chartScript": {
        "script": "X",
        "variables": [
          "X"
        ]
      }
    }
  ],
  "edges": [
    {
      "from": "t",
      "to": "__M_X_2"
    },
    {
      "from": "__M_X_2",
      "to": "__equals_1"
    },
    {
      "from": "e",
      "to": "__power_4"
    },
    {
      "from": "t",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "X",
      "to": "__multiply_5",
      "semantic": "direct",
      "weight": 1.0
    },
    {
      "from": "__multiply_5",
      "to": "__power_4",
      "role": "exp"
    },
    {
      "from": "__power_4",
      "to": "__E_3"
    },
    {
      "from": "__E_3",
      "to": "__equals_1"
    }
  ],
  "classification": {
    "kind": "algebraic"
  }
}

flowchart RL
  classDef scalar fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef vector fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef constant fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef number fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef expression fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef text fill:#1b3a1e,stroke:#66bb6a,color:#c8e6c9,font-size:15px
  classDef operator fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef function fill:#1a2740,stroke:#42a5f5,color:#bbdefb,font-size:15px
  classDef relation fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef result fill:#332a0b,stroke:#ffa726,color:#ffe0b2,font-size:15px
  classDef annotation fill:#131a2c,stroke:#4a5580,color:#b0bcd0,font-size:15px
  __equals_1{"="}:::relation
  __M_X_2{{"$M_{X}(\cdot)$"}}:::function
  t["$t$"]:::scalar
  __E_3{{"$E(\cdot)$"}}:::function
  __power_4(("$(\cdot)^{\cdot}$")):::operator
  e["$e$"]:::scalar
  __multiply_5(("$\times$")):::operator
  X["$X$"]:::scalar
  t --> __M_X_2
  __M_X_2 --> __equals_1
  e --> __power_4
  t --> __multiply_5
  X --> __multiply_5
  __multiply_5 -->|exp| __power_4
  __power_4 --> __E_3
  __E_3 --> __equals_1
  linkStyle 0 stroke:#aaa,stroke-width:2px
  linkStyle 1 stroke:#aaa,stroke-width:2px
  linkStyle 2 stroke:#aaa,stroke-width:2px
  linkStyle 3 stroke:#ef5350,stroke-width:3px
  linkStyle 4 stroke:#ef5350,stroke-width:3px
  linkStyle 5 stroke:#aaa,stroke-width:2px
  linkStyle 6 stroke:#aaa,stroke-width:2px
  linkStyle 7 stroke:#aaa,stroke-width:2px