AMS310 Summer2011 hw5-ch5&6&7 solutions

AMS310 Summer2011 hw5-ch5&6&7 solutions -...

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Unformatted text preview: Homework 5 (Chapter 5&6&7) Solutions 1. Given the joint pmf f(x1,x2): (a) Find f(100,100) f(100,100)=0.05 (b) Find P(X1>=100, X2<=100) 35 . 15 . 05 . 05 . 10 . ) 100 , 250 ( ) , 250 ( ) 100 , 100 ( ) , 100 ( ) 100 , 100 ( 2 1 = + + + = + + + = ≤ ≥ f f f f X X P (c) Find the marginal distribution for X1 = = = = w o x x x x f . , 250 , 5 . 100 , 2 . 50 , 3 . ) ( 1 1 1 1 1 i.e. 35 . 05 . 05 . 10 . ) 200 , 100 ( ) 100 , 100 ( ) , 100 ( ) , ( ) 100 ( 2 2 1 1 = + + = + + = = ∑ f f f x x f f x (d) Find P(X1<=100) 5 . 2 . 3 . ) 100 ( ) 50 ( ) 100 ( 1 1 1 = + = + = ≤ f f X p (e) Check if X1 and X2 are independent f(x1,x2) x2 100 200 x1 50 .10 .05 .15 100 .10 .05 .05 250 .05 .15 .30 ) 200 , 50 ( 15 . 5 . * 3 . ) 200 ( ) 50 ( 2 1 f f f = = = But ) 100 , 50 ( 05 . 075 . 25 . * 3 . ) 100 ( ) 50 ( 2 1 f f f = ≠ = = 2 1 & x x ∴ are not indep. 2. ≤ ≤ ≤ ≤ + = . . 1 , 1 ) ( 5 6 ) , ( 2 w o y x y x y x f (a) Find the marginal distribution of Y. ∫ ∫ ∞ ∞- ≤ ≤ + = + = + = + = = 1 2 2 2 2 2 . , 1 , 5 3 5 6 ) 2 1 ( 5 6 1 ) 2 ( 5 6 ) ( 5 6 ) , ( ) ( w o y y y xy x dx y x dx y x f Y f y (b) Find P(1/4<=Y<=3/4) 3. X1 & X2 are independent. X1~exponential ( 1 λ...
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This note was uploaded on 09/14/2011 for the course AMS 310 taught by Professor Mendell during the Summer '08 term at SUNY Stony Brook.

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AMS310 Summer2011 hw5-ch5&amp;6&amp;7 solutions -...

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