# # 常用概率分布

## # 伯努利分布

$X$ $0$ $1$
$P$ $q$ $p$

$P(X=x)=p^{x}(1-p)^{1-x}=\left\{\begin{array}{ll}p & \text { if } \; x=1 \\ q & \text { if } \; x=0\end{array}\right.$

$\mathbb{E}(X) = \sum_{i=0}^1 x_i P(X=x) = 0 + p = p$

$\text{Var}[X]=\sum_{i=0}^{1}\left(x_{i}-E[X]\right)^{2} P(X=x)=(0-p)^{2}(1-p)+(1-p)^{2} p=p(1-p)=p q$

Tips

## # 二项分布

$f(k, n, p)=\text{Pr}(k ; n, p)=\text{Pr}(X=k)=\left(\begin{array}{l}n \\ k\end{array}\right) p^{k}(1-p)^{n-k}$

$\left(\begin{array}{l}n \\ k\end{array}\right)=\frac{n !}{k !(n-k) !}$

$\mathbb{E}(X_1 + X_2) = \mathbb{E}(X_1) + \mathbb{E}(X_2)$

$\text{Var}(X_1 + X_2) = \text{Var}(X_1) + \text{Var}(X_2)$

$X = \sum_{i=0}^n X_i$

$\mathbb{E}(X) = \sum_{i=1}^n \mathbb{E}(X_i) = np$

$\text{Var}(X) = \sum_{i=1}^n \text{Var}(X_i) = np(1-p)$

## # 一维高斯分布

$p_{X}(x)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right)$

$h(X)=\frac{1}{2}\left[1+\log \left(2 \pi \sigma^{2}\right)\right]$

1. Wikipedia: Binomial distribution (opens new window)