ポアソン分布の確率質量関数から，ポアソン確率変数の期待値(平均) E[X]，分散 V[X]，標準偏差 √V[X] を計算する方法を示します．一般に，離散確率変数の期待値は，確率質量関数とその引数の積の総和として定義されます．
ポアソン分布の期待値・分散・標準偏差
ポアソン分布(Poisson distribution)の確率質量関数(probability mass function; PMF)
(1) 
が与えられたとき，その期待値(expected value)![E[X]](https://k-san.link/wp-content/ql-cache/quicklatex.com-4249815a79caf7890371efa1a609bb7f_l3.png "Rendered by QuickLaTeX.com")
，分散(variance)![V[X]](https://k-san.link/wp-content/ql-cache/quicklatex.com-0d5508ee65fbf6af600d78e8cb19d1b4_l3.png "Rendered by QuickLaTeX.com")
，標準偏差(standard deviation)![\sqrt{V[X]}](https://k-san.link/wp-content/ql-cache/quicklatex.com-3f52bc5237eab53e22134f90855afaa6_l3.png "Rendered by QuickLaTeX.com")
は，それぞれ
(2) ![\begin{equation*} E[X] = \lambda \end{equation*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-e0223c64ef39a23893d6c231b1093070_l3.png "Rendered by QuickLaTeX.com")
(3) ![\begin{equation*} V[X] = \lambda \end{equation*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-efe50e2b97f09da194b06e43fc9ad53c_l3.png "Rendered by QuickLaTeX.com")
(4) ![\begin{equation*} \sqrt{V[X]} = \sqrt{\lambda} \end{equation*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-f600745b839168ef6db82b3ea5d67cc8_l3.png "Rendered by QuickLaTeX.com")
となる．□
関連ページ
指数分布の計算：期待値(平均)，分散，標準偏差の求め方【確率論】
正規分布（ガウス分布）の計算：期待値(平均)，分散，標準偏差の求め方【確率論】
ポアソン分布の期待値の計算・求め方
(5) ![\begin{eqnarray*} E[X] &=& \sum_{k=0}^{\infty} k \; P_k \\ &=& \sum_{k=0}^{\infty} k \frac{\lambda^k \; e^{-\lambda} }{k!} \\ &=& \sum_{k=1}^{\infty} k \frac{\lambda^k \; e^{-\lambda} }{k!} \\ &=& \sum_{k=1}^{\infty} \frac{\lambda^k \; e^{-\lambda} }{(k-1)!} \\ &=& \lambda \sum_{k=1}^{\infty} \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} \\ &=& \lambda \sum_{k'=0}^{\infty} \frac{\lambda^{k'} \; e^{-\lambda} }{k'!} \quad (k':=k-1)\\ &=& \lambda \cdot 1 \\ &=& \lambda \end{eqnarray*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-72bebb2e360c44359f7fabee55abe844_l3.png "Rendered by QuickLaTeX.com")
ポアソン分布の分散と標準偏差の計算・求め方
(6) ![\begin{eqnarray*} V[X] &:=& E\left( (X - E[X])^2 \right) \\ &=& E\left( X^2 - 2XE[X] + E[X]^2 \right) \\ &=& E[X^2] - E\left( 2XE[X] \right) + E\left( E[X]^2 \right) \\ &=& E[X^2] - 2E[X]E[X] + E[X]^2 \\ &=& E[X^2] - E[X]^2 \end{eqnarray*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-246042e46efb69ff85a3ee1b5726733e_l3.png "Rendered by QuickLaTeX.com")
![E[X^2]](https://k-san.link/wp-content/ql-cache/quicklatex.com-d4ed6d18fd14734e28e532b101ed5a87_l3.png "Rendered by QuickLaTeX.com")
と![E[X]^2](https://k-san.link/wp-content/ql-cache/quicklatex.com-f48feacc9b601273846fbf29b1e4d38b_l3.png "Rendered by QuickLaTeX.com")
を計算すればよい． (7) ![\begin{eqnarray*} E[X^2] &=& \sum_{k=0}^{\infty} k^2 \; P_k \\ &=& \sum_{k=0}^{\infty} k^2 \frac{\lambda^k \; e^{-\lambda} }{k!} \\ &=& \sum_{k=1}^{\infty} k^2 \frac{\lambda^k \; e^{-\lambda} }{k!} \\ &=& \sum_{k=1}^{\infty} k \frac{\lambda^k \; e^{-\lambda} }{(k-1)!} \\ &=& \sum_{k=1}^{\infty} \lambda k \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} \\ &=& \sum_{k=1}^{\infty} \lambda (k-1) \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} + \sum_{k=1}^{\infty} \lambda \cdot 1 \cdot \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} \\ &=& \sum_{k=1}^{\infty} \lambda (k-1) \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} + \sum_{k=2}^{\infty} \lambda \cdot 1 \cdot \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} \\ &=& \sum_{k=2}^{\infty} \lambda^2 \frac{\lambda^(k-2) \; e^{-\lambda} }{(k-2)!} + \sum_{k=1}^{\infty} \lambda \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} \\ &=& \lambda^2 \sum_{k''=0}^{\infty} {k''}^2 \frac{\lambda^{k''} \; e^{-\lambda} }{{k''}!} + \lambda \sum_{k'=0}^{\infty} {k'}^2 \frac{\lambda^{k'} \; e^{-\lambda} }{{k'}!} \quad(k'':=k-2, k':= k-1)\\ &=& \lambda^2 + \lambda \end{eqnarray*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-5370a1aa3e41b67bf625b71493f57278_l3.png "Rendered by QuickLaTeX.com")
である．また(5)式より
(8) ![\begin{equation*} E[X]^2 = \lambda^2 \end{equation*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-e406eff19f13f80d10e53a606119210f_l3.png "Rendered by QuickLaTeX.com")
である．これらを(6)式に代入すれば，ポアソン分布の分散
(9) ![\begin{eqnarray*} V[X] &=& E[X^2] - E[X]^2 \\ &=& \lambda^2 + \lambda - \lambda^2 \\ &=& \lambda \end{eqnarray*}](https://k-san.link/wp-content/ql-cache/quicklatex.com-554f227c260ec58fd88b15ee46e08536_l3.png "Rendered by QuickLaTeX.com")
を得る．
また，標準偏差は分散の正平方根なので
(10)
となる．
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