概率(英文)随机变量(1)

1、随机现象中,有很多问题会与数值发生联系,如产品检验中,我们关心的是抽样中出现的废品数,车间供电问题中某时刻工作的机床数,电话问题中某段时间的话务量.有些看起来与数值无关的随机现象也常常能联系数值来描述.如抛硬币,我们可以把正反面分别用数字1和0来表示,要计算n次实验中正面出现的次数,只要计算 “1”出现的次数.一般地,A为某随机事件,可通过示性函数与数值发生联系:0,1,AAIA若 不发生若 发生将随机实验的结果数量化,就是我们要讨论的随机变量.1 Random Variables and Discrete DistributionsExample 1 There are three ball

2、s in a bag, the ball are labeled by number1,2,3.Now take two balls from the bag with replacement, consider the sum of the numbers of the balls.( , ) ,1,2,3the sample spaceis Si j i jIf we let X represent the sum of the two numbers, then for everyelement in S, there is a value x correspond to it, and

3、 it is easy to seeX can be 2,3,4,5,6.Example 2 Toss a coin three times ,let x represent the times that thehead occurred ,then ,SHHH HHT HTH THH HTT THT TTH TTTWhere “H” means the head and “T” means the tail.Thus the values of X are 3,2,2,2,1,1,1,0 ,respectively.1. Definition: A real-valued function

4、defined on the space S is called a random variable Denoted as X, Y, ZceX:., 0, 1)(whiteeredeeXAbout random variables:是单值函数,每个样本点对应唯一一个数x.一般用大写字母X,Y,Z等表示随机变量,小写字母x,y等表示实数.随机变量的取值是随机的,它取每个值的概率由样本空间中样本点的情况定.4.若L是一实数集,随机变量X在L上取值,则( ),( )( )XLBe X eLP XLP BP e X eL表示事件所以2. Discrete Distributions If X can

5、 take a finite number different valuesor at most an infinite sequence of different valueskxx,1,21kxxxThen X is said to be a discrete random variable.Or X has a discrete distributions .There are a few expressions of distribution about X as following : probability function (p.f.) 概率函数概率函数 f (x) = P( X

6、 = x )kkpP Xx或者分布律：See Figure 3.1 ( P 57 );, 2 , 1, 0)1( kpk. 1)2(1 kkpnnpppxxxX2121)2(Xkpnxxx21nppp21分布率Several district distribution 1.Bernoulli distribution: X can only take two values: 0and 1P58 (也叫做两点分布或者(0-1)分布)kp0p 11pX2. The Uniform Distribution on Integers (P58)otherwisekxkxf, 0, 2 , 1,1)(3

7、. The Binomial DistributionExample : a coin is tossed n times .kp0p 11piXSolution : (1) Every time (2) To sum up ( n times )Let X be the number of times that head appears Then X takes n+1 different integers,0,1,kn knP Xkp qknk Noted as : X B (n, p) X is called to have the binomial distribution with

8、parameters n and p . X 0 1 k n PIn fact :nXXXX21Moreover , X is called as :N重重 Bernoulli Distributions,0,1,2,( )0,xn xnp qxnf xxotherwise p.f.(分布函数)：(1),0,1,kkn knP XkC ppkn分布律：12,.niXXXXX其中服从两点分布二项分布应用很广泛.例例 按规定,某种型号的电子元件的使用寿命超过1500小时为一级品,已知某大批产品中的一级品率为0.2,现在从中随机地抽查20只,问20只元件中恰有k (k=0,1,2.n)只为一级品的概

9、率为多少?说明:这是不放回抽样,但是元件数量很大,而抽取的元件数相对很少,检查20只元件相当于做20次伯努力实验,用X表示20只中一级品数,则(20,0.2)XbThe Poisson Distribution（结果为可列无限个数 :0，1，2，) P193).(.tan0, 2 , 1 , 0,!eXasNotedtconsaiskkkXPkChecking the normal property 1 主要两方面的随机现象服从泊松分布1.社会生活中:电话台收到的呼叫次数,公交车站的乘客人数等.2.物理学领域:放射性物质经过某区域的质点数,显微镜下的微生物数等.5. Geometric Dis

10、tribution tossing a coin until a head appears 1 2 k P 1/2 1/4 k2/1Jacob BernoulliBorn: 27 Dec 1654 in Basel, SwitzerlandDied: 16 Aug 1705 in Basel, Switzerland(伯努利资料)2 Continuous Distribution Definition: (P61) X can assume every value in an interval . If there exists a nonnegative Function f such th

11、atbadxxfbXaP)(f (x) is called the probability density function of X .2. The property of f (x) :(1) f (x) 0 for all x .(2) 1)(dxxfExample3.2.2 : otherwisexcxxfIf, 040,)(Then : ( 1 ) c=? ( 2 ) P -2 X 2 = ? (4)12?PXxo)(xf11d)( xxfS1SxxfSxxd)(211 1x 2x 3.Geometric Illustration : ( see Fig ) p is the tot

12、al area under the curve f (x) .Note 1. PX=a = 0 2.Nonuniqueness of the p.d.f. 4. A few continuous distributions :(1) The Uniform Distribution on an interval X assume every value in the interval S= a, b in a random way .X is uniformly distributed in the interval S= a, b otherwisebxaabxf, 0,1)(xo)(xf

13、a b(2) The Exponential Distribution 0, 00,)(xxexfxP Xst XsP Xt指数分布的无记忆性：指数分布在可靠性理论与排队论中有广泛应用。(3) The Normal Distribution（5.5）).,(.tan)0(,e21)(22)(22NXasNotedtsconsarexxfx,e21)(22 xxx The Standard Normal Distribution ( )1, 03 The Distribution Function ( All have a common characterization )1. Definiti

14、on （d.f.) （P67）F ( x ) = P X x for all x Rx)(xXPxF 1x2x21xXxP)()(12xFxF0,1,1,12,4( )3,23,41,3.xxF xxx .32,2523,21Pr)(XPXPXPofobabilitytheandxFtheFindXkp321 412141Example 1 :-123xF ( x)1/43/41.Example 2 : ( the F (x) of the Exponential Distribution ). 0 , 0, 0,e1)(x-xxxF2. Basic Property of F (x) : P

15、67)()(,) 1 (2121xFxFthenxxIf. 1)(lim, 0)(lim)2(xFxFxx)()()3(xFxF3. Theorems : P 69 (1) P X x = 1 F (x)()()2(1221xFxFxXxP21)(xxdxxfThe relation of f (x) and F (x) :(1) For Discrete Distributions xo)(xF 1x 2x 1p 2p 1)()(xxkkxXPxF(2) For Continuous Distributions dxxdFxfdttfxFx)()()()(. 0 , 0, 0,e1)(x-xxxFExample ：If th