概率论与数理统计思维导图

离散型样本空间：个数有限、可列个

连续性样本空间：个数不可列、无限个

∅：不可能事件

Ω：必然事件

基本事件：事件的结果不可再分

包含关系：事件A发生,事件B必然发生,记作A ⊆B

相等关系,记作A=B

互不相容：事件A与B不可能同时发生

事件的并(和)：事件A与B至少有一个发生,记住A∪B

事件的交(积)：事件A与B同时发生,记住A∩B、AB

事件的差：事件A发生而B不发生的概率，记住A-B

对立事件：事件A的对立事件，A与B互为对立事件的充要条件A∩B=∅，A∪B=Ω

交换律

结合律

分配律

对偶律

非负性：P(A)>0

正则性：P(Ω)=1

可列可加性

排列

组合

有限个样本

等可能性(每个样本点发生的可能性相等)

公式：p(A)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{事件A所含样本点个数}{Ω中所有样本点的个数}"><span></span><span></span></span>

公式：P(A)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{S_{A}}{S_Ω}"><span></span><span></span></span>

样本空间无限

P(∅)=0

有限可加性

若A⊇B，则P(A-B)=P(A)-P(B)推论：若A⊇B，则P(A)＞P(B)

P(A-B)=P(A)-P(AB)

P(A∪B)=P(A)+P(B)-P(AB)推论：P(A∪B)≤P(A)+P(B)

离散随机变量

连续随机变量

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>,&nbsp; &nbsp;在离散场合<br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<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)]&nbsp; &nbsp;&nbsp;

离散随机变量

连续随机变量

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

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)

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

P(X=k)=<span class="equation-text" data-index="0" data-equation="C^n_k" contenteditable="false"><span></span><span></span></span>&nbsp;<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)

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)

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

P(X=k)=&nbsp;<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{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>

E(X)=μ

Var(X)=<span class="equation-text" contenteditable="false" data-index="0" data-equation="{σ^2}"><span></span><span></span></span>

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

离散型

连续型