式展開: 二項分布(binomial distribution)の積率母凾数(moment-generating function)と期待値(expected value)・分散(variance)

\begin{array}{rcl} X & \sim & B (n, p) \\ f (X = x) & = & {\begin{cases} _{n} C_{x} p^{x} (1 - p)^{n - x} & x \in {0, 1, 2, \dots, n} \\ 0 & x \notin {0, 1, 2, \dots, n} \end{cases} \dots 確 率 密 度 凾 数 \end{array}

\begin{array}{rcl} M_{X} (t) & = & E [e^{t x}] \\ = & \sum_{k = 0}^{n} e^{t x}_{n} C_{x} p^{x} {(1 - p)}^{n - x} \\ = & \sum_{k = 0}^{n}_{n} C_{x} {(e^{t} p)}^{x} {(1 - p)}^{n - x} \\ = & _{n} C_{0} {(e^{t} p)}^{0} {(1 - p)}^{n - 0} +_{n} C_{1} {(e^{t} p)}^{1} {(1 - p)}^{n - 1} +_{n} C_{2} {(e^{t} p)}^{2} {(1 - p)}^{n - 2} + \dots +_{n} C_{n} {(e^{t} p)}^{n} {(1 - p)}^{n - n} \\ = & {e^{t} p + (1 - p)}^{n} \\ \dots (A + B)^{D} =_{D} C_{0} A^{0} B^{D - 0} +_{D} C_{1} A^{1} B^{D - 1} + \dots +_{D} C_{D} A^{D} B^{D - D} = \sum_{k = 0}^{D}_{D} C_{k} A^{k} B^{D - k} (二 項 関 係) \\ = & {(e^{t} p - p + 1)}^{n} \end{array}

\begin{array}{rcl} E [X] & = & M_{X}^{(1)} (t) \\ = & {\frac{d M_{X} (t)}{d t} |}_{t = 0} \\ = & {\frac{d}{d t} {(e^{t} p - p + 1)}^{n} |}_{t = 0} \\ = & {\frac{d}{d u} u^{n} \frac{d u}{d t} |}_{t = 0} \dots u = e^{t} p - p + 1, \frac{d u}{d t} = \frac{d}{d t} (e^{t} p - p + 1) = e^{t} p \\ = & {n u^{n - 1} e^{t} p |}_{t = 0} \\ = & {n {(e^{t} p - p + 1)}^{n - 1} e^{t} p |}_{t = 0} \\ = & {n p e^{t} {(e^{t} p - p + 1)}^{n - 1} |}_{t = 0} \\ = & n p e^{0} {(e^{0} p - p + 1)}^{n - 1} \\ = & n p \cdot 1 {(1 \cdot p - p + 1)}^{n - 1} \\ = & n p {(1)}^{n - 1} \\ = & n p \dots X の 期 待 値 の 定 義 か ら 求 め た 二 項 分 布 の 期 待 値 と 同 じ \end{array}

\begin{array}{rcl} E [X^{2}] & = & M_{X}^{(2)} (t) \\ = & {\frac{d^{2} M_{X} (t)}{d t^{2}} |}_{t = 0} \\ = & {\frac{d^{2}}{d t^{2}} {(e^{t} p - p + 1)}^{n} |}_{t = 0} \\ = & {\frac{d}{d t} n p e^{t} {(e^{t} p - p + 1)}^{n - 1} |}_{t = 0} \\ = & {n p \frac{d}{d t} e^{t} {(e^{t} p - p + 1)}^{n - 1} |}_{t = 0} \dots \frac{d}{d x} c f (x) = c \frac{d}{d x} f (x) (c : 定 数) \\ = & {n p \frac{d}{d t} u v |}_{t = 0} \dots u = e^{t}, v = {(e^{t} p - p + 1)}^{n - 1} \\ = & {n p {(\frac{d}{d t} u) v + u (\frac{d}{d t} v)} |}_{t = 0} \\ = & {n p [(e^{t}) v + u {(n - 1) {(e^{t} p - p + 1)}^{n - 2} p e^{t}}] |}_{t = 0} \dots \frac{d u}{d t} = e^{t}, \frac{d v}{d t} = (n - 1) {(e^{t} p - p + 1)}^{n - 2} p e^{t} \\ = & {n p [(e^{t}) {(e^{t} p - p + 1)}^{n - 1} + e^{t} {(n - 1) {(e^{t} p - p + 1)}^{n - 2} p e^{t}}] |}_{t = 0} \dots u = e^{t}, v = {(e^{t} p - p + 1)}^{n - 1} \\ = & {n p e^{t} {(e^{t} p - p + 1)}^{n - 1} + n (n - 1) p^{2} e^{2 t} {(e^{t} p - p + 1)}^{n - 2} |}_{t = 0} \\ = & n p e^{0} {(e^{0} p - p + 1)}^{n - 1} + n (n - 1) p^{2} e^{2 \cdot 0} {(e^{0} p - p + 1)}^{n - 2} \\ = & n p \cdot 1 \cdot {(1 \cdot p - p + 1)}^{n - 1} + n (n - 1) p^{2} \cdot 1 \cdot {(1 \cdot p - p + 1)}^{n - 2} \\ = & n p {(1)}^{n - 1} + n (n - 1) p^{2} {(1)}^{n - 2} \\ = & n p + n (n - 1) p^{2} \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} \\ = & M_{X}^{(2)} (t) - {(M_{X}^{(1)} (t))}^{2} \\ = & n p + n (n - 1) p^{2} - (n p)^{2} \\ = & n p + n^{2} p^{2} - n p^{2} - n^{2} p^{2} \\ = & n p - n p^{2} \\ = & n p (1 - p) \dots X (X - 1) の 期 待 値 を 利 用 し た 二 項 分 布 の 分 散 と 同 じ \end{array}

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