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09_18ans - STAT 410 Examples for Fall 2009 Fact Let X and Y...

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STAT 410 Examples for 09/18/2009 Fall 2009 Fact : Let X and Y be continuous random variables with joint p.d.f. ( y x f , . Then ( 29 ( 29 - = - + dx x w x f w f , Y X ( 29 ( 29 - = - + dy y y w f w f , Y X (convolution) Proof : ( 29 ( 29 - - - + = dx dy y x f w x w , Y X F . let u = y + x , then d u = d y , y = u x , , w x w ( 29 ( 29 - - - + = dx du x u x f w w , Y X F = ( 29 - - - w du dx x u x f , . ( w f Y X + = ( 29 w ' Y X F = ( 29 - - dx x w x f , . Fact : Let X and Y be independent continuous random variables. Then ( 29 ( 29 ( 29 - = - + dx x w f x f w f Y X Y X ( 29 ( 29 ( 29 - = - + dy y f y w f w f Y X Y X
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1&2. a) Let X and Y be two independent Exponential random variables with mean 1. Find the probability distribution of Z = X + Y. That is, find ( z f Z = ( z f Y X + . ( 29 - = otherwise 0 0 X x x e x f ( 29 < - - + - + - = = otherwise 0 otherwise 0 0 Y w x e x w e x w f x w x w ( w f Y X + = ( 29 ( 29 - - dx x w f x f Y X = + - - w x w x dx e e 0 = - w w dx e 0 = w e w - , w > 0. 3. When a person applies for citizenship in Neverland, first he/she must wait X years for an interview, and then Y more years for the oath ceremony. Thus the total wait is W = X + Y years. Suppose that X and Y are independent, the p.d.f. of X is f X ( x ) = 2 / x 3 , x > 1, zero otherwise, and Y has a Uniform distribution on interval ( 0, 1 ) . Find the p.d.f.
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