# Hw03 - STAT 410 Homework#3(due Friday September 19 by 3:00...

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STAT 410 Fall 2008 Homework #3 (due Friday, September 19, by 3:00 p.m.) 1. Suppose that the random variables X and Y have joint p.d.f. f ( x , y ) given by f ( x , y ) = C x 2 y , 0 < x < y , x + y < 2. a) Sketch the support of ( X , Y ) . b) What must the value of C be so that f ( x , y ) is a valid joint p.d.f. ? c) Find P ( X + Y < 1 ) . 2. Suppose that the random variables X and Y have joint p.d.f. f ( x , y ) given by f ( x , y ) = 6 x 2 y , 0 < x < y , x + y < 2. a) Find the marginal probability density function for X. b) Find the marginal probability density function for Y. 3. Suppose that ( X, Y ) is uniformly distributed over the region defined by 0 y 1 – x 2 and – 1 x 1. a) What is the joint probability density function of X and Y ? b) Find the marginal densities of X and Y. c) Find the two conditional densities. From the textbook: 2.1.6 Let f ( x , y ) = e x y , 0 < x < , 0 < y < , zero elsewhere, be the pdf of X and Y . Then if Z = X + Y , compute P ( Z 0 ) , P ( Z 6 ) , and, more generally, P ( Z z ) , for 0 < z < . What is the pdf of Z ?

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2.1.7 Let X and Y have the pdf f ( x , y ) = 1, 0 < x < 1, 0 < y < 1, zero elsewhere. Find the
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