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# Hw04 - STAT 410(due Friday September 26 by 3:00 p.m...

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STAT 410 Fall 2008 Homework #4 (due Friday, September 26, by 3:00 p.m.) 1. Let X and Y have the joint probability density function f X, Y ( x , y ) = ° ± ° ² ³ < < < + otherwise 0 1 0 4 x y y x a) Find f Y ( y ) . b) Find f Y | X ( y | x ) . c) Find E ( Y | X ) . d) Are X and Y independent? Justify your answer. e) Find Cov ( X, Y ) . From the textbook: 2.3.3 Let f ( x 1 , x 2 ) = 3 2 2 1 21 x x , 0 < x 1 < x 2 < 1, zero elsewhere, be the joint pdf of X 1 and X 2 . (a) Find the conditional mean and variance of X 1 , given X 2 = x 2 , 0 < x 2 < 1. (b) Find the distribution of Y = E ( X 1 | X 2 ) . (c) Determine E ( Y ) and Var ( Y ) and compare these to E ( X 1 ) and Var ( X 1 ) , respectively. 2.3.10 Let X 1 and X 2 have joint pmf p ( x 1 , x 2 ) described as follows: ( x 1 , x 2 ) ( 0, 0 ) ( 0, 1 ) ( 1, 0 ) ( 1, 1 ) ( 2, 0 ) ( 2, 1 ) p ( x 1 , x 2 ) 18 1 18 3 18 4 18 3 18 6 18 1 and p ( x 1 , x 2 ) is equal to zero elsewhere. Find the two marginal probability mass functions and the two conditional means. Hint: Write the probabilities in a rectangular array.

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2.3.11 Let us choose at random a point from interval ( 0, 1 ) and let the random variable X 1 be equal to the number which corresponds to that point. Then choose a point at random
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