概率

概率

离散型概率分布

$P(X=x)=P_i(i=1,2,3...,n)$

$$\begin{gathered} 1.0\le p_i\le 1 \\ 2.{\sum }_{i=1}^n{p}_i=1 \\ 3.P(a\le b)={\sum }_{x=a}^{b}P(X=x) \end{gathered}$$

期望

$E(X) =\sum_{i=1}^{n}{x_{i}p_{i}} $

方差

$$\begin{gathered} V(X)=E[(X-m{)}^2]={\sum }_{i=1}^n(x_i-m{)}^2{p}_i \\ V(X)=E[(x-m{)}^2]=E(x^2)-E(x{)}^2 \end{gathered}$$

$$\begin{gathered} V(ax+b)=a^{2}V(x) \\ \sigma (ax+b)=|a|\sigma (x) \end{gathered}$$

二项分布 B(n, p)

$$\begin{gathered} 期望：E(x)=np \\ 方差：V(x)=np(1-q) \\ 标准差：\sigma (x)=\sqrt{np(1-q)} \end{gathered}$$

二项式定理

$(a+b{)}^n={\sum }_{i=1}^n{C}_n^r{a}^r{b}^{n-r}$

n次独立重复试验概率：
$P(X=r)=C_{i=n}^r{p}^r(1-p{)}^r$