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# hw4 - UCSD ECE153 Prof Young-Han Kim Handout#15 Thursday...

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UCSD ECE153 Handout #15 Prof. Young-Han Kim Thursday, April 21, 2011 Homework Set #4 Due: Thursday, April 28, 2010 1. Two independent uniform random variables. Let X and Y be independently and uni- formly drawn from the interval [0 , 1]. (a) Find the pdf of U = max( X, Y ) . (b) Find the pdf of V = min( X, Y ). (c) Find the pdf of W = U - V . (d) Find the probability P {| X - Y | ≥ 1 / 2 } . 2. Waiting time at the bank. Consider a bank with two tellers. The service times for the tellers are independent exponentially distributed random variables X 1 Exp( λ 1 ) and X 2 Exp( λ 2 ), respectively. You arrive at the bank and ±nd that both tellers are busy but that nobody else is waiting to be served. You are served by the ±rst available teller once he/she becomes free. Let the random variable Y denote your waiting time. Find the pdf of Y . 3. Two envelopes. An amount A is placed in one envelope and the amount 2 A is placed in another envelope. The amount A is ±xed but unknown to you. The envelopes are shu²ed and you are given one of the envelopes at random. Let X denote the amount you observe in this envelope. Designate by Y the amount in the other envelope. Thus ( X, Y ) = b ( A, 2 A ) , with probability 1 2 , (2 A, A ) , with probability 1 2 . You may keep the envelope you are given, or you can switch envelopes and receive the

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hw4 - UCSD ECE153 Prof Young-Han Kim Handout#15 Thursday...

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