概率论笔记

01分布

$$P{X = k} = p^k(1-p)^{1-k}, k=0,1$$

$$E(X) = p$$

$D(X) = p(1 - p)$

二项分布 $X\sim b(n, p)$

$$P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}, k = 0, 1, 2\cdots n.$$

$E(X) = n p$

$D(X) = np(1 - p) $

泊松分布 $X\sim \pi(\lambda)$

$$P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!}, k = 0, 1, 2\cdots$$

$$E(X) = \lambda$$

$$D(X) = \lambda$$

均匀分布 $X\sim U(a, b)$

$$f(x)=\begin{cases}\frac{1}{b-a},&a<x<b \0,&else \end{cases}$$
$$E(X)=\frac{a+b}{2}$$
$$D(X)=\frac{(b-a)^2}{12}$$
$$F(X)=\begin{cases}0,&x<a\\frac{x-a}{b-a},&a\le x<b\1,&x\ge b\end{cases}$$

指数分布

$$f(x)=\begin{cases}\frac{1}{\theta}e^{-x/\theta},&x>0 \0,&else \end{cases}$$

$$E(X) = \theta$$

$$D(X) = \theta^2$$

$$F(X)=\begin{cases}1-e^{-x/\theta},&x>0\0,&else\end{cases}$$

正态分布 $X\sim (\mu, \sigma^2)$

$f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}, -\infty<x<+\infty$

$F(X) = \frac{1}{\sqrt{2\pi}\sigma}\int^x_{-\infty}e^{-\frac{(t-\mu)^2}{2\sigma^2}}\text{d} t$

$E(X) = \mu$

$D(X) = \sigma^2$

$F(X) = P(X \le x) = \Phi(\frac{x-\mu}{\sigma})$