概率论第二章

2017-03-13 13:51:44   0  举报

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第二章主要探讨了离散型随机变量及其概率分布。首先，我们介绍了离散型随机变量的定义和性质，包括其可能取值的有限性和可列举性。接着，我们详细讨论了几种常见的离散型概率分布，包括二项分布、泊松分布和几何分布等。每种分布都有其特定的参数和概率质量函数，通过这些函数，我们可以计算出任意取值的概率。此外，我们还介绍了期望值和方差的概念，它们是描述随机变量特性的重要统计量。最后，我们通过一些实际问题，展示了如何利用这些理论知识来解决实际问题。这一章的内容为后续的学习打下了坚实的基础。

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大纲/内容

随机变量

定义：设随机试验的样本空间为S=｛e｝，X=X(e)是定义在样本空间S上的实值单值函数。称X=X(e)为随机变量。

表示方法：以大写字母X,Y,Z,W…表示随机变量，以小写x,y,z,w……表示实数

例：一般，若L是一个实数集合，将X在L上取值写成｛X∈L｝。它表示事件B=｛e|X(e)∈L｝，此时有P｛X∈L｝=P(B)=P｛e|X(e)∈L｝

离散型随机变量及其分布律

定义1：有些随机变量全部可能的取值是有限个或可列无限多个，这种随机变量叫做离散型随机变量

定义2：设离散型随机变量X所以可能的取值为xk(k=1,2，……)，X取各个可能值的概率，即事件｛X=xk｝的概率为P｛X=xk｝=pk，k=1,2……(分布律)

条件：(1)Pk≥0 n=1,2,…<div>　　    (2)∑pk=1</div>

三种重要离散型随机变量

（0-1）分布

定义：0—1分布是n=1情况下的二项分布。即只先进行一次事件试验，该事件发生的概率为p,不发生的概率为q=1-p。这是一个最简单的分布，任何一个只有两种结果的随机现象。

分布律两种写法

P｛X=k｝=p^k (1-q)^(1-k),k=0,1 (0<p<1)

见书 P33

伯努利，二项分布

定义：伯努利试验是在同样的条件下重复地、相互独立地进行的一种随机试验。假设该项试验独立重复地进行了n次，那么我们就称这一系列重复独立的随机试验为n重伯努利试验，或称为伯努利概型。

伯努利实验特点：该随机试验只有两种可能结果——发生或者不发生。重复是指每次试验中P(A)=p保持不变；独立是指各次试验结果不相互影响。

伯努利定理：设在一次试验中，事件A发生的概率为p（0<p<1），则在n重伯努利试验中，事件A恰好发生k次的概率为b(k;n,p)=<img src="http://e.hiphotos.baidu.com/baike/s%3D201/sign=adbb02f548c2d562f608d7edd61090f3/a71ea8d3fd1f41348389a9eb221f95cad0c85eec.jpg"><div>推论：设在一次试验中，事件A首次发生的概率为p(0<p<1)，则在伯努利试验序列中，事件A在第k次试验中才首次发生的概率为p(1-p)^k-1(k=1,2...)</div>

二项分布：一般地在n次独立重复试验中，用ξ表示事件A发生的次数，如果事件发生的概率是P,则不发生的概率 q=1-p，<div>N次独立重复试验中发生K次的概率是</div><div>P(ξ=K)=  (K=0,1,2,3,…n）</div><div>那么就说ξ服从二项分布。.其中P称为成功概率。</div><div>记作：ξ~B(n,p)</div><div>期望：Eξ=np</div><div>方差：Dξ=npq</div>

泊松分布

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随机变量的分布函数

定义：设X是一个随机变量，x是任意实数，函数F（x）=P｛X=<x｝,-∞<x<∞称为X的分布函数

分布函数性质

1、F（x）是一个不减函数

2、0=<F(x)<=1   且F(-∞)=limF(x)=0  F(∞)=limF(x)=1

3、F(x+0)=F(x),即F(x)是右连续的

连续型随机变量及其概率密度

定义

如果对于随机变量X的分布函数F（x）,存在非负的可积函数f(x)使对于任意实数x有公式（4.1)则称X为连续型随机变量，f(x)称为X的概率密度函数，简称概率密度

性质

f(x)>=o

f(x)在R上的定积分为1