随机变量及其分布
数学期望
离散随机变量
E(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum_{i=1}^{∞}{x_{i}f(x_{i})}"><span></span><span></span></span>
连续随机变量
E(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="∫_{-∞}^{∞}{xf(x)dx}"><span></span><span></span></span>
E(aX)=aE(X)
E[g(X)]=<span class="equation-text" data-index="0" data-equation="\sum_{i}{g(x_{i})f(x_{i})}" contenteditable="false"><span></span><span></span></span>, 在离散场合<br> <span class="equation-text" contenteditable="false" data-index="1" data-equation="∫_{-∞}^{∞}{g(x)f(x)dx}"><span></span><span></span></span>，在连续场合
E(c)=c
E[g(X)+h(X)]=E[g(X)]+E[h(X)]
离散随机变量的概率分布列
表格形式
非负性：每个概率都大于等于0
正则性：概率之和为1
连续随机变量的概率密度函数
f(x)是非负可积函数，x∈(-∞，+∞)
P(a＜X≤b)=∫(a,b)f(x)dx，记作x~f(x)
非负性：f(x)＞0，正则性：f(x)在-∞到+∞上的积分等于1
<span class="equation-text" data-index="0" data-equation="{F(x)}=" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="{∫^{x}_{-∞}{f(t)d(t)}}" contenteditable="false"><span></span><span></span></span>,且<span class="equation-text" contenteditable="false" data-index="2" data-equation="{F`(x)=f(x)}"><span></span><span></span></span>
P(a≤x≤b)=P(a＜x＜b)=P(a≤x＜b)=P(a＜x≤b)
离散随机变量及其分布<br>X~<span class="equation-text" data-index="0" data-equation="P_i" contenteditable="false"><span></span><span></span></span>,则F(X)= <span class="equation-text" contenteditable="false" data-index="1" data-equation="\sum_{x_i<=x}(P_i)"><span></span><span></span></span> <br>
定义：随机变量仅可能取有限个或可列个值
0-1分布
P(X=x)=<span class="equation-text" data-index="0" data-equation="p^x" contenteditable="false"><span></span><span></span></span>(1-<span class="equation-text" contenteditable="false" data-index="1" data-equation="p^{1-x}"><span></span><span></span></span>)，x=0，1
二项分布
P(X=k)=<span class="equation-text" data-index="0" data-equation="C^n_k" contenteditable="false"><span></span><span></span></span> <span class="equation-text" data-index="1" data-equation="p^k" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="2" data-equation="(1-p)^{n-k}"><span></span><span></span></span><br>X~B(n,p)
E(X)=np
Var(X)=np(1-p)
泊松分布
P(X=k)=<span class="equation-text" data-index="0" data-equation="\frac{λ^k}{k!}" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="e^{-λ}"><span></span><span></span></span>,k=0,1,2,……，λ＞0，记为：X~P(λ)
E(X)=λ
Var(X)=λ
超几何分布
P(X=k)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{C^k_M C^{n-k}_{N-M}}{C^n_N}"><span></span><span></span></span>,k=0,1,2……，r
E(X)=n <span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{M}{N}"><span></span><span></span></span>
Var(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{nM(N-M)(N-n)}{N^2(N-1)}"><span></span><span></span></span>
几何分布
P(X=k)= <span class="equation-text" data-index="0" data-equation="(1-p)^{k-1}{p}" contenteditable="false"><span></span><span></span></span>，k=1，2……
E(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{p}"><span></span><span></span></span>
Var(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1-p}{p^2}"><span></span><span></span></span>
无记忆性
连续随机变量及其分布<br>X~<span class="equation-text" contenteditable="false" data-index="0" data-equation="{f(x)}"><span></span><span></span></span>，则F(X)=<span class="equation-text" data-index="1" data-equation="∫^{-∞}_{x}{f(t)dt}" contenteditable="false"><span></span><span></span></span>
定义：随机变量的可能取值充满数轴上一个区间(a,b)
均匀分布<br>X~U(a,b)
E(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{a+b}{2}"><span></span><span></span></span>
Var(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{(b-a)^{2}}{12}"><span></span><span></span></span>
正态分布<br>X~(μ，<span class="equation-text" contenteditable="false" data-index="0" data-equation="{σ^2}"><span></span><span></span></span>)
E(X)=μ
Var(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="{σ^2}"><span></span><span></span></span>
指数分布<br>X~<span class="equation-text" contenteditable="false" data-index="0" data-equation="{Exp}"><span></span><span></span></span>(λ)
E(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{λ}"><span></span><span></span></span>
Var(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{λ^2}"><span></span><span></span></span>
无记忆性
伽马分布
贝塔分布
分布函数
概念：F(x)=P(X≤x)，记为X~F(x)，也可表示FX(x) 把X写成F的下标,x∈(-∞，+∞)
F(x)是分布函数，是累计概率，是事件{X≤x}的所有概率之和
单调性：单调递增
有界性：0≤F(x)≤1且F(-∞)=0，F(+∞)=1
右连续性：F(a+0)=F(a)
计算分布函数
离散型
“符合范围，直接相加”(0-1分布，二项分布，泊松分布)
P(x＞b)=1-F(b)
P(a＜x≤b)=F(b)-F(a)
