二维随机变量的联合分布多维随机变量.PPT

Chapter 3 Random vectors and their numerical characteristics Bivariate distribution Continuous joint distributions and density functions Marginal distribution for discrete distribution Marginal density function § 3.3 CONDITIONAL DISTRIBUTION 3.4 Functions of random vectorsFunctions of discrete random vectors Example 3.23 Suppose that is independent of with marginal density and respectively, then the density function of is specified by  Remark Under the conditions of Example 3.23, one can easily find the density function of is Remark (1) If (X,Y) is continuously distributed, then a sufficient condition for X and Y to be independent is f(x,y)=fX(x)fY(y) (2) If (X,Y) is discrete distributed with law Pi,j=P{X=xi,Y=yj},i,j=1,2,...then a sufficient and continously for X and Y to be independent is Pi,j=Pi?.P?j EX：try to determine that whether the (X,Y) in Example1，Exmaple2，Example3 are independent or not? Example 4. Suppose that the d.f. of (X,Y) is give by the following chart and that X is independent of Y, try to determine a and b. Definition 3.7 Suppose is an discrete two-dimensional distribution, for given , if, then the conditional distribution law for given can be represented by which is defined as (3.14) and (1) (2) Definition 3.9 Suppose that for any , holds, if exist, then the limit is called the conditional distribution of for given and denoted it by or . Theorem 3.6 Suppose that has continuous density function and then (3.15) is a continuous d.f. and its density function