几何分布
定义
\[P(X = k) = q^{k - 1}p, \quad k = 1,2,..., \ 0 < p < 1, \ q = 1 - p,
\]
记为 \(X \sim G(p)\).
期望
\[EX = \frac{1}{p}.
\]
证明
\[EX = \sum_{k = 1}^{\infty }kq^{k - 1}p = p\sum_{k = 1}^{\infty }kq^{k - 1} = p\sum_{k = 1}^{\infty }\frac{dq^{k}}{dq} = p \cdot \frac{d\displaystyle \sum_{k = 1}^{\infty }q^{k}}{dq} = p \cdot \frac{d \displaystyle \frac{q}{1 - q}}{dq} = \frac{p}{(1 - q)^2},
\]
所以,
\[EX = \frac{1}{p}.
\]
方差
\[DX = \frac{1 - p}{p^2}.
\]
