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 = psum_{k = 1}^{infty }kq^{k - 1} = psum_{k = 1}^{infty }frac{dq^{k}}{dq} = p cdot frac{ddisplaystyle 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}.
]