hw3 - UCSD ECE153 Handout #12 Prof. Young-Han Kim Thursday,...

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Unformatted text preview: UCSD ECE153 Handout #12 Prof. Young-Han Kim Thursday, April 14, 2011 Homework Set #3 Due: Thursday, April 21, 2011 1. Time until the n-th arrival. Let the random variable N ( t ) be the number of packets arriving during time (0 ,t ]. Suppose N ( t ) is Poisson with pmf p N ( n ) = ( λt ) n n ! e- λt for n = 0 , 1 , 2 ,.... Let the random variable Y be the time to get the n-th packet. Find the pdf of Y . 2. Diamond distribution. Consider the random variables X and Y with the joint pdf f X,Y ( x,y ) = braceleftbigg c, if | x | + | y | ≤ 1 / √ 2 , , otherwise , where c is a constant. (a) Find c . (b) Find f X ( x ) and f X | Y ( x | y ). (c) Are X and Y independent random variables? Justify your answer. 3. First available teller. 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 find that both tellers are busy but that nobody else is waiting to be served. You are served by the first available teller once he/she is free. What is the probability that you are served by the first teller? 4. Coin with random bias. You are given a coin but are not told what its bias (probability of heads) is. You are told instead that the bias is the outcome of a random variable P ∼ U[0 , 1]. To get more infromation about the coin bias, you flip it independently 10 times. Let X be the number of heads you get. Thus X ∼ Binom(10 ,P ). Assuming that X = 9, find and sketch the a posteriori probability of P , i.e., f P | X ( p | 9)....
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This note was uploaded on 07/07/2011 for the course EECS 153 taught by Professor Kim during the Spring '11 term at UCSD.

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hw3 - UCSD ECE153 Handout #12 Prof. Young-Han Kim Thursday,...

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