隨機(jī)變量基礎(chǔ)精選教育課件

1、 本次課內(nèi)容概率的基本術(shù)語(yǔ)隨機(jī)變量的定義隨機(jī)變量的分布函數(shù)與概率密度多維隨機(jī)變量 第一章 隨機(jī)變量基礎(chǔ)1.1 概率的基本術(shù)語(yǔ)隨機(jī)試驗(yàn) Random Experiment隨機(jī)事件 Random Event基本事件 Elementary (Simple) Event樣本空間 Sample Space頻率 Frequency概率 Probability投擲骰子出現(xiàn)1點(diǎn)1，2，3，4，5，6投擲骰子出現(xiàn)偶數(shù)點(diǎn)樣本空間隨機(jī)事件基本事件關(guān)于樣本空間的注釋：Discrete Sample Space:Toss a die: S=1,2,3,4,5,6連續(xù)的樣本空間:由多次子試驗(yàn)構(gòu)成的樣本空間:看下例Toss

2、 a coin：S=Head, Tail=H,TIF 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 isS=HHH, HHT, HTH, HTT, THH, THT, TTH, TTTThe event “ one head and two tails” is defined byE=HTT,

3、 THT, TTH由多次子試驗(yàn)構(gòu)成的樣本空間可數(shù)無(wú)窮的樣本空間S=S1 S1 =HH, HT, TH, TT, S1=H,T利用頻率估計(jì)概率n次重復(fù)試驗(yàn)中，事件A發(fā)生的次數(shù)為nA，比值稱為事件A發(fā)生的頻率。頻率反映了事件A發(fā)生的頻繁程度，若事件A發(fā)生的可能性大，那么相應(yīng)的頻率也大，反之則較小。 概率計(jì)算機(jī)模擬：投擲一枚均勻硬幣，模擬計(jì)算出現(xiàn)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)

4、=0; % tail endnumber=number+x(i,1);% count number of headsendP=number/N;1.2 隨機(jī)變量的定義(Definition of a random variable)設(shè)隨機(jī)試驗(yàn)E的樣本空間為S=e，如果對(duì)于每一個(gè)eS，有一個(gè)實(shí)數(shù)X(e)與之對(duì)應(yīng)，這樣就得到一個(gè)定義在S上的單值函數(shù)X(e)，稱X(e)為隨機(jī)變量，簡(jiǎn)記為X。 隨機(jī)變量是定義在樣本空間S上的單值函數(shù)1. 定義Interpretation of random variable:SeReal lineRandom variable is a function that a

5、ssigns a numerical value to the outcome of the experiment.A coin tossSe1Real line10e2Mapping of the outcome of a coin toss into the set of real numberA discrete random variable is a random variable that can be take on at most a countable number of possible values根據(jù)隨機(jī)變量取值的不同可以分為： 連續(xù)型隨機(jī)變量(Continuous r

6、andom variable) 離散型隨機(jī)變量(Discrete random variable)2. 概率分布列Xx1x2.xnpkp1p2.pnProbability mass function (PMF)(0,1)分布 指示型隨機(jī)變量隨機(jī)變量的可能取值為0和1兩個(gè)值，PMF為PMF:0 1(0,1)分布的隨機(jī)變量；指示型隨機(jī)變量；貝努里隨機(jī)變量；Bernoulli random variableLet A be an event of interest in some experiment, e.g., a device is not defective. We say that a “

7、success” occurs if A occurs when we perform the experiment.Bernoulli random variable IA is equal to 1 if A occurs and zero otherwise. 例：信息傳輸問(wèn)題（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

8、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 variableExample: Transmission error in a binary communications channel .Let

9、X be the number of errors in n independent transmissions. Find the PMF of x. Find the probability of one or fewer errors01011-1-X takes on values in the set 0,1,nIf there is no error, each transmission results in a 0If there is an error, each transmission results in a 1The probability of k errors in

10、 n bits transmissions is given as followsWe call X the binomial random variable貝努里試驗(yàn) 隨機(jī)試驗(yàn)只有兩種結(jié)果。 產(chǎn)品合格或不合格 元件沒(méi)有失效或失效 信息發(fā)送成功或失敗 打靶命中或脫靶 目標(biāo)檢測(cè)或漏檢貝努里序列（Bernoulli Sequence）N次連續(xù)的獨(dú)立的貝努里試驗(yàn)稱為貝努里序列Consecutive independent Bernoulli trials comprise a Bernoulli sequence.貝努里試驗(yàn)-N重貝努里試驗(yàn)(0,1)分布-二項(xiàng)式分布信息傳輸泊松分布(Poisson

11、distribution)1.3 分布函數(shù)和概率密度函數(shù)Probability Density Function, （PDF） Distribution Function or Cumulative Distribution Function, (CDF)1. 定義2. 分布函數(shù)的性質(zhì)（Properties of the CDF）分布函數(shù)是右連續(xù)的不減函數(shù)，在負(fù)無(wú)窮處為零，正無(wú)窮處為1。對(duì)于連續(xù)型隨機(jī)變量，取某一特定值的概率是為零的。即PX=x=0對(duì)于離散型隨機(jī)變量，分布函數(shù)為階梯函數(shù)。概率密度對(duì)于離散型隨機(jī)變量，它的概率密度函數(shù)是一串函數(shù)之和。 Check YourselfSuppose X

12、=c, Where c is constantWhich of following is correct?A. B. C. D. E. None of Above Check YourselfSuppose X=c, Where c is constantWhich of following is correct?A. B. C. D. E. None of Above3. 常見(jiàn)概率分布 正態(tài)分布（Normal），也稱高斯（Gauss）分布 -4-3-2-10123400.10.20.30.40.50.60.70.8N(0,1)正態(tài)分布概率密度 標(biāo)準(zhǔn)正態(tài)分布函數(shù)瑞利分布（Rayleigh）瑞利

13、分布概率密度2 02468101200.050.10.150.20.250.30.350.4指數(shù)（Exponential）分布指數(shù)分布概率密度 0123456700.511.5指數(shù)分布可以用來(lái)表示獨(dú)立隨機(jī)事件發(fā)生的時(shí)間間隔，比如，旅客進(jìn)機(jī)場(chǎng)的時(shí)間間隔，維基（wikipedia)新條目出現(xiàn)的時(shí)間間隔，電話呼叫的時(shí)間間隔等，是一個(gè)率（Rate)參數(shù)。 對(duì)數(shù)正態(tài)分布（LogNormal）高分辨率雷達(dá)雜波分布01234567891000.10.20.30.40.5對(duì)數(shù)正態(tài)分布概率密度 為尺度參數(shù)為形狀參數(shù)1.4 多維隨機(jī)變量及其分布 Multiple Random Variables and Dist

14、ributions 1. 定義Se比如：同時(shí)投擲一個(gè)1元硬幣和一個(gè)1角硬幣，樣本空間為S=HH,TH,TT,HT。HHTHTTHTxy2. 二維分布函數(shù)和概率密度 Bivariate CDF and PDF 二維分布函數(shù)圖解 定義：性質(zhì)：參看教材二維概率密度： 對(duì)于二維離散隨機(jī)變量，定義聯(lián)合概率質(zhì)量函數(shù)（Join PMF)二維概率密度：3. 條件分布(Conditional Distribution) 條件分布函數(shù)條件概率密度稱隨機(jī)變量X、Y獨(dú)立條件概率質(zhì)量函數(shù)（Conditional PMF):以隨機(jī)事件為條件的概率密度:幾個(gè)重要的貝葉斯公式:全概率公式:貝葉斯公式:概率分布與概率密度的全概率公式:Example: Communication Channel with Discrete Input and Continuous OutputThe 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 -