Chapter 3 Multivariate Random Variables 多元随机变量：3章多随机变量多元随机变量

1、chapter 3 multivariate random variables1.n-dimensional variables n random variables x1，x2，.,xn compose a n-dimensional vector (x1,x2,.,xn), and the vector is named n-dimensional variables or random vector.2. joint distribution of random vectordefine function f(x1,x2,xn)= p(x1x1,x2 x2,.,xn xn)the joi

2、nt distribution function of random vector (x1,x2,.,xn).3.1 two-dimensional random variablesdefinition 3.1-p53 let (x, y) be 2-dimensional random variables. define f(x,y)=px x, y ythe bivariate cdf of (x, y) . the joint cdf for two random variables 00,yyxxyx geometric interpretation：the value of f( x

3、, y) assume the probability that the random points belong to area in dark for (x1, y1), (x2, y2) r2, (x1x2，y1y2 ), then px1x x2，y1y y2 f(x2, y2)f(x1, y2) f (x2, y1)f (x1, y1).(x1, y1)(x2, y2)(x2, y1)(x1, y2)1x2x3x1y2y3ysuppose that the joint cdf of (x,y) is f(x,y), find the probability that (x,y) st

4、ands in area g .answer2133233112231322(, ) (,)(,)(,)(,) ( ,)(,)( ,)(,)px ygf xyf xyf xyf xyf x yf xyf x yf xy ljoint distribution f(x, y) has the following characteristics:0),(lim),( yxffyx1),(lim),( yxffyx0),(lim),( yxfyfx0),(lim),( yxfxfy(1) for all (x, y) r2 , 0 f(x, y) 1, (2) monotonically incre

5、ment for any fixed y r, x1x2 yields f(x1, y) f(x2 , y)； for any fixed x r, y1y2 yields f(x, y1) f(x , y2). );y,x(f)y, x(flim)y, 0 x(f0 xx00 ).y, x(f)y, x(flim)0y, x(f0yy00 (3) right continuous for x r, y r, (4) for all (x1, y1), (x2, y2) r2, (x1x2，y1y2 ), f(x2, y2)f(x1, y2) f (x2, y1)f (x1, y1) 0.co

6、nversely, any real-valued function satisfied the aforementioned 4 characteristics must be a joint distribution function of 2-dimensional variables.example 1. let the joint distribution of (x,y) is)3()2(),(yarctgcxarctgbayxf 1) find the value of a，b，c。2) find p0x2,0yy211010 xdydxyxp11xyfind (1)the value of a； (2) the value of f(1,1)； (3) the probability of (x, y) stand in region d：x 0, y 0, 2x+3y 6 casesother for , 00, 0,),(),()32(yxaeyxfyxyxanswer (1) since6 a 101032)32()1)(1(6)1 ,