式展開: 確率母凾数 ( probability generating function )

非負の整数値をとる離散型確率変数

X

に対して以下のように確率母凾数(probability generating function;積率母凾数ではない)が定義される．

\begin{array}{rcl} G_{X} (t) & = & E [t^{X}] \dots 積 率 母 凾 数 は M_{X} (t) = E [e^{t X}] \\ = & \sum_{k} t^{k} P (X = k) \\ \dots P (X) : 確 率 質 量 凾 数, \sum_{k} : X の 定 義 範 囲 す べ て の k で の 和 \end{array}

\begin{array}{rcl} {G_{X}^{(1)} (t) |}_{t = 1} & = & {\frac{d}{d t} G_{X} (t) |}_{t = 1} \\ = & {\sum_{k \geq 1} k t^{k - 1} P (X = k) |}_{t = 1} \\ = & \sum_{k \geq 1} k 1^{k - 1} P (X = k) \\ = & \sum_{k \geq 1} k P (X = k) \\ = & E [X] \end{array}

\begin{array}{rcl} {G_{X}^{(2)} (t) |}_{t = 1} & = & {\frac{d^{2}}{d t^{2}} G_{X} (t) |}_{t = 1} \\ = & {\frac{d}{d t} \sum_{k \geq 1} k t^{k - 1} P (X = k) |}_{t = 1} \\ = & {\sum_{k \geq 2} k (k - 1) t^{k - 2} P (X = k) |}_{t = 1} \\ = & \sum_{k \geq 2} k (k - 1) 1^{k - 2} P (X = k) \\ = & \sum_{k \geq 2} k (k - 1) P (X = k) \\ = & E [X (X - 1)] \end{array}

\begin{array}{rcl} V [X] & = & E [{(X - E [X])}^{2}] \\ = & E [X^{2}] - E {[X]}^{2} \dots E [{(X - E [X])}^{2}] = E [X^{2}] - {[X]}^{2} \\ = & E [X^{2}] - E [X] + E [X] - E {[X]}^{2} \\ = & E [X^{2} - X] + E [X] - E {[X]}^{2} \dots E [X \pm Y] = E [X] \pm E [Y] \\ = & E [X (X - 1)] + E [X] - E {[X]}^{2} \\ = & ({G_{X}^{(2)} (t) |}_{t = 1}) + ({G_{X}^{(1)} (t) |}_{t = 1}) - {({G_{X}^{(1)} (t) |}_{t = 1})}^{2} \end{array}

\begin{array}{rcl} G_{Z} (t) & = & E [t^{Z}] \\ = & E [t^{X + Y}] \\ = & E [t^{X} t^{Y}] \dots A^{B + C} = A^{B} A^{C} \\ = & E [t^{X}] E [t^{Y}] \dots E [A B] = E [A] E [B] A と B が 独 立 の 場 合 \\ = & G_{X} (t) G_{Y} (t) \end{array}

式展開