常见概率分布的概述

Discrete

Bernoulli distribution

• 概率质量函数

• P(X=x) = f_X(x) =\begin{cases}(1-p)^{1-x}p^x & \text{当x取值为0或1时}\\ 0 & \text{其他情况下}\end{cases}

  • expectation
    • E(X) = p

Binomial distribution

该概率质量函数（PMF）定义为：
P(X = k) = f_X(k) = \begin{cases} \dbinom{n}{k} p^k (1 - p)^{n - k}, & \text{其中 } k = 0, 1, ..., n \\ 0, & \text{其他情况} \end{cases}

• expectation

  • E(X) = np

• variance

  • var(X) = np(1-p)

Geometric distribution

• pmf
  • $f_X(k) = P(X= k) =\left{\begin{aligned}p(1-p)^{k-1} & \quad \text{for k=1,2,3...}\ 0 & \quad