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§3.5 Distribution of Special Functions

§3.5 Distribution of Special Functions. Sum and difference Z=X+Y, Z=X-Y Product and quotient Z=XY, Z=YX. Functions of discrete random vectors. Suppose that (X, Y) ～ P(X ＝ x i , Y ＝ y j ) ＝ p ij ， i, j ＝ 1, 2, …

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§3.5 Distribution of Special Functions

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  1. §3.5 Distribution of Special Functions • Sum and difference Z=X+Y, Z=X-YProduct and quotientZ=XY, Z=Y\X

  2. Functions of discrete random vectors Suppose that (X, Y)～P(X＝xi, Y＝yj)＝pij ，i, j＝1, 2, … then Z＝g(X, Y)～P{Z＝zk}＝ ＝pk , k＝1, 2, … or

  3. EX Suppose that X and Y are independent, and the pmfs of X and Y are Find the pmf of Z=X+Y. Solution Because X and Y are independent, so

  4. then

  5. EXSuppose that X and Y are independent and both are uniformly distributed on 0-1 with law X01 • P q p • Try to determine the distribution law of • W＝X＋Y ;(2) V＝max(X, Y)； • (3) U＝min(X, Y); • (4)The joint distribution law of w and V .

  6. 2 0 1 1 0 1 1 1 0 0 0 1 W 0 1 2 V 0 1 0 0 0

  7. Example Suppose and are independent of each other, then ExampleSuppose and are independent of each other , then

  8. Example Suppose and are independent of each other, then Proof i = 0 , 1 , 2 , … j = 0 , 1 , 2 , … then

  9. r = 0 , 1 , … then

  10. a) Sum and difference Let X and Y be continuous r.v.s with joint pdf f (x, y), Let Z=X+Y ,Find the pdf of Z=X+Y. The cdf of Z is : D={(x, y): x+y ≤z}

  11. Let x=u-y, then

  12. Then the pdf of Z=X+Y is : Or

  13. In particular, if X and Y are independent, then which is called the convolution of fx(.) and fy(.).

  14. Example 1 If X and Y are independent with the same pdf Find the pdf of Z=X+Y. Solution i.e.

  15. So when or , when, when , so

  16. Example 2 If X and Y are independent with the same distributionN(0,1) , Find the pdf of Z=X+Y. Solution

  17. let then Z=X+Y ～N(0,2).

  18. let f(x,y) is the joint pdf of (X,Y), then the pdf of Z=Y\X is When X and Y are independent, 二、Z=Y\X, Z=XY

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