概率论笔记

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

$$E(X) = p$$

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

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

$E(X) = n p$

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

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

$$E(X) = lambda$$

$$D(X) = lambda$$

$$f(x)=begin{cases}frac{1}{b-a},&a<x<b ,&else end{cases}$$
$$E(X)=frac{a+b}{2}$$
$$D(X)=frac{(b-a)^2}{12}$$
$$F(X)=begin{cases}0,&x<a\frac{x-a}{b-a},&ale x<b1,&xge bend{cases}$$

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

$$E(X) = theta$$

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

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

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

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

$E(X) = mu$

$D(X) = sigma^2$

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