HW5 - X(b The manufacturer sells the article for a fixed...

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OR 3500/5500, Summer’09, Chen Homework 5 Due on Wednesday, June 3, 3:00pm. Problem 1 In a population on size 100, 20 are smoker. A random sample of size 5 is chosen without replacement. Let X denote the number of smokers in the sample. Compute Var( X ). Hint: Define suitable indicator random variables. Problem 2 Let X and Y have a joint pdf given by f X,Y ( x,y ) = 1 if 0 < y < 1 , y - 1 < x < 1 - y 0 otherwise . (a) Find Cov( X,Y ) and ρ X,Y . Are X and Y independent? (b) Find Cov( | X | , | Y | ) and ρ | X | , | Y | . (c) Find Cov(3 | X | , 4 | Y | ) and ρ 3 | X | , 4 | Y | . (d) Find Var(3 X - 2 Y ). Problem 3 A random variable X , which represents the weight (in ounces) of an article, has density function given by f X ( x ) = x - 8 for 8 x 9 10 - x for 9 < x 10 0 otherwise . (a) Calculate the mean and variance of the random variable
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Unformatted text preview: X (b) The manufacturer sells the article for a fixed price of $2 . 00. He guarantees to refund the purchase money to any customer who finds the weight of his article to be less than 8 . 25 oz. His cost of production is related to the weight of the article by x/ 15 + . 35. Find the expected profit per article and its standard deviation. The following problem is optional for 3500 students, mandatory for students enrolled in 5500 Problem 4 Let p i = P ( X = i ) and suppose that p 1 + p 2 + p 3 = 1. If EX = 2, what values of p 1 ,p 2 and p 3 ( a ) maximize and ( b ) minimize Var X ? 1...
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.

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