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3.4 Independence of random vectors

3.4 Independence of random vectors. Definition It is said that X is independent of Y for any real number a<b, c<d,p{a<X  b,c<Y  d} =p{a<X  b}p{c<Y  d},i.e. event{a<X  b}is independent of {c<Y  d}.

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3.4 Independence of random vectors

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  1. 3.4 Independence of random vectors Definition It is said that X is independent of Y for any real number a<b, c<d,p{a<Xb,c<Yd} =p{a<Xb}p{c<Yd},i.e. event{a<Xb}is independent of {c<Yd}. Definition3.10A sufficient and necessary condition for random variables X and Y to be independent is F(x,y)=FX(x)FY(y)

  2. Remark----P61 (1) 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 (2) 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)

  3. Example 1. 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. Example 2----P62Example3.8

  4. Note: Joint distribution Marginal distribution Joint distribution Conditional distribution Example1 Suppose that X is independent of Y ,Given the pmf of X and Y : try to determine the joint pmf of (X,Y).

  5. Solution X is independent of Y→ the joint pmf of (X,Y) is

  6. Example2 Suppose that X and Y are independent,Given the pdf of X and Y : (1)try to determine the joint pdf of (X,Y). (2) Solution (1)Because X and Y are independent, so

  7. EX:P69---10 Homework:P69---13,14

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