第二讲随机变量基础.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,本次课内容,概率的基本术语,随机变量的定义,随机变量的分布函数与概率密度,多维随机变量,第一章 随机变量基础,1.1 概率的基本术语,随机试验,Random Experiment,随机事件,Random Event,基本事件,Elementary(Simple)Event,样本空间,Sample Space,频率,Frequency,概率,Probability,投掷骰子出现1点,1，2，3，4，5，6,投掷骰子出现偶数点,样本空间,随机事件,基本事件,关于样本空间的注释：,Discrete Sample Space:,Toss a die:S=1,2,3,4,5,6,连续的样本空间:,由多次子试验构成的样本空间:看下例,Toss a coin：S=Head,Tail=H,T,IF we toss a coin three times and let the triplet xyz denote the outcome“x on the first toss,y on the second toss,z on the third toss”,then the sample space of the experiment is,S=HHH,HHT,HTH,HTT,THH,THT,TTH,TTT,The event“one head and two tails”is defined byE=HTT,THT,TTH,计算机模拟：,投掷一枚均匀硬币，模拟计算出现正面的概率。,number=0;,for i=1:N,%set up simulation for 4 coin toses,if rand(1,1)0.5,%toss coin with p=0.5,x(i,1)=1;,%head,else,x(i,1)=0;,%tail,end,number=number+x(i,1,);,%count number of heads,end,P=number/N;,1.2 随机变量的定义,(Definition of a random variable),设随机试验E的样本空间为S=e，如果对于每一个e,S,，有一个实数X(e)与之对应，这样就得到一个定义在S上的单值函数X(e)，称X(e)为随机变量，简记为X。,随机变量是定义在样本空间S上的单值函数,1.定义,Interpretation of random variable:,S,e,Real line,Random variable is a,function,that assigns a numerical value to the outcome of the experiment.,A coin toss,S,e,1,Real line,1,0,e,2,Mapping of the outcome of a coin toss into the set of real number,A discrete random variable,is a random variable that can be take on at most a countable number of possible values,根据随机变量取值的不同可以分为：,连续型随机变量(Continuous random variable),离散型随机变量(Discrete random variable),2.概率分布列,X,x,1,x,2,.,x,n,p,k,p,1,p,2,.,p,n,Probability mass function(PMF),(0,1),分布,指示型随机变量,随机变量的可能取值为,0,和,1,两个值，,PMF,为,PMF:,0 1,(0,1)分布的随机变量；,指示型随机变量；,贝努里随机变量；,Bernoulli random variable,Let A be an event of interest in some experiment,e.g.,a device is not defective.We say that a“success”occurs if A occurs when we perform the experiment.,Bernoulli random variable I,A,is equal to 1 if A occurs and zero otherwise.,例：信息传输问题（Message Transmissions）,Let X be the number of times needs to be transmitted until it arrivers correctly at its destination.Find the probability that X is an a even number.,X is a discrete random variable taking on values from S=1,2,3,.,The event X=k occurs if k-1 consecutive erroneous transmissions(failures)followed by a error-free one(success),X is called the,geometric random variable,Example:,Transmission error in a binary communications channel.,Let X be the number of errors in n independent transmissions.Find the PMF of x.Find the probability of one or fewer errors,0,1,0,1,1-,1-,X takes on values in the set 0,1,n,If there is no error,each transmission results in a 0,If there is an error,each transmission results in a 1,The probability of k errors in n bits transmissions is given as follows,We call X the,binomial random variable,贝努里试验 随机试验只有两种结果。,产品合格或不合格,元件没有失效或失效,信息发送成功或失败,打靶命中或脱靶,目标检测或漏检,贝努里序列（Bernoulli Sequence）,N次连续的独立的贝努里试验称为贝努里序列,Consecutive independent Bernoulli trials comprise a Bernoulli sequence.,贝努里试验-N重贝努里试验,(0,1)分布-二项式分布,信息传输,泊松分布,(Poisson distribution),1.3 分布函数和概率密度函数,Probability Density Function,（PDF）,Distribution Function,or Cumulative Distribution Function,(CDF),1.定义,2.分布函数的性质（Properties of the CDF）,分布函数是右连续的不减函数，在负无穷处为零，正无穷处为1。对于连续型随机变量，取某一特定值的概率是为零的。即PX=x=0,对于离散型随机变量，分布函数为阶梯函数。,概率密度,对于离散型随机变量，它的概率密度函数是一串,函数之和。,Check Yourself,Suppose X=c,Where c is constant,Which of following is correct?,A.,B.,C.,D.,E.None of Above,Check Yourself,Suppose X=c,Where c is constant,Which of following is correct?,A.,B.,C.,D.,E.None of Above,3.常见概率分布,正态分布（Normal），也称高斯（Gauss）分布,-4,-3,-2,-1,0,1,2,3,4,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,N(0,1)正态分布概率密度,标准正态分布函数,瑞利分布（Rayleigh）,瑞利分布概率密度,2,0,2,4,6,8,10,12,0,0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,指数（Exponential）分布,指数分布概率密度,0,1,2,3,4,5,6,7,0,0.5,1,1.5,指数分布可以用来表示独立随机事件发生的时间间隔，比如，旅客进机场的时间间隔，维基（wikipedia)新条目出现的时间间隔，电话呼叫的时间间隔等，,是一个率（Rate)参数。,对数正态分布（LogNormal）,高分辨率雷达杂波分布,0,1,2,3,4,5,6,7,8,9,10,0,0.1,0.2,0.3,0.4,0.5,对数正态分布概率密度,为尺度参数,为形状参数,1.4 多维随机变量及其分布,Multiple Random Variables and Distributions,1.定义,S,e,比如：同时投掷一个1元硬币和一个1角硬币，样本空间为S=HH,TH,TT,HT。,HH,TH,TT,HT,x,y,2.二维分布函数和概率密度,Bivariate CDF and PDF,二维分布函数图解,定义：,性质：参看教材,二维概率密度：,对于二维离散随机变量，,定义联合概率质量函数（Join PMF),二维概率密度：,3.条件分布(Conditional Distribution),条件分布函数,条件概率密度,称随机变量X、Y独立,条件概率质量函数（Conditional PMF):,以随机事件为条件的概率密度:,几个重要的贝叶斯公式:,全概率公式:,贝叶斯公式:,概率分布与概率密度的全概率公式:,Example:Communication Channel with Discrete Input and Continuous Output,The input X to a communication channel is+1 volt or-1 volt.The output Y of the channel is the input plus a noise voltage N that is uniformly distributed in the interval from-2 volts to+2 volts.,Find PX=+1,Y0,Solution:,1/2,When the input X=1,the output Y is uniformly distributed in the interval,Therefore,本次课小结：,（1）概率的基本术语：,随机试验 基本事件 随机事件 样本空间，频率与概率,（2）随机变量的定义,从样本空间到实轴的映射,（3）随机变量的分布,PMF CDF PDF,典型随机变量的分布,（4）条件分布,