HW8_Problem_Set

# HW8_Problem_Set - working time 3[5 3 2 = 10 pts A...

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ORIE 3500/5500 Fall Term 2008 Assignment 8 Note that the due date is Monday, November 10 , at 1:00 pm you need to mention your section number on the top of your answer script. 1. [5 + 5 = 10 pts] In a computer lab there is a printer which starts working at 9 a.m. and goes down after T hours, where T exp(1 / 4). The time to repair it is R hours, where R exp(1). Assume that the printer can go down at most once in a day and the repairing starts immediately after it goes down. What is the probability that the printer is working (a) at 1 p.m. (b) between 1 and 2 p.m.? 2. [3 + 3 + 4 = 10 pts] The number of jobs for an employee has Poisson (10) distribution. Each job requires a time exp(2) hours independent of other jobs and the number of jobs. Find the transform associated to total working time for the employee. Use it to ﬁnd the mean and variance of the total
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Unformatted text preview: working time. 3. [5 + 3 + 2 = 10 pts] A transmitter sends out a signal X ∼ N (1 , 1 / 4). If x is the original signal, then because of some noise the receiver gets a signal Y which is normal with mean a + bx and standard deviation b | x | for some constants a and b such that b > 0. If it is known that E ( Y ) = 0 , var ( Y ) = 3, then (a) ﬁnd a and b , (b) cov ( X, Y ) and (c) correlation coeﬃcient ρ ( X, Y ). 4. [4 + 6 = 10 pts] Suppose that X, Y have joint PDF f X,Y ( x, y ) = ± 8 xy, ≤ x ≤ y ≤ 1 otherwise . (a) Find var ( XY ) and (b) correlation coeﬃcient ρ ( X, XY ). 5. [5 + 5 = 10 pts] If X, Y are independent Uniform (0 , 1) random variables, then ﬁnd the correlation coeﬃcients (a) ρ ( X, X + Y ) and (b) ρ ( X-Y, X + Y ). 1...
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## This homework help was uploaded on 02/20/2009 for the course ORIE 3500 taught by Professor Weber during the Fall '08 term at Cornell.

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