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homework11

homework11 - V AR T 4(2 pts(Textbook exercise 4.39 Let X Y...

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Probability for Engineering Applications Assignment #11, due at the beginning of class on Thursday, April 20 th , 2006 1. (2 pts) (Textbook exercise 4.31) The random vector variable ( X, Y ) has the joint pdf f X,Y ( x, y ) = k ( x + y ) 0 < x < 1 , 0 < y < 1 Find f Y ( y | x ) and f X ( x | y ). 2. (2 pts) (Textbook exercise 4.37) The number of defects in a VLSI chip is a Poisson random variable with rate r . However, r is itself a gamma random variable with parameters α and λ . a. Find the pmf for N , the number of defects. b. Use conditional expectation to find E [ N ] and V AR [ N ]. 3. (2 pts) (Textbook exercise 4.35) A customer entering a store is served by clerk i with prob- ability p i , i = 1 , 2 , · · · , n . The time taken by clerk i to service a customer is an exponentially distributed random variable with parameter α i . a. Find the pdf of T , the time taken to service a customer. b. Find
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Unformatted text preview: V AR [ T ]. 4. (2 pts) (Textbook exercise 4.39) Let X , Y , and Z have joint pdf f X,Y,Z ( x, y, z ) = k ( x + y + z ) ≤ x ≤ 1 , ≤ y ≤ 1 , ≤ z ≤ 1 , a. Find k . b. Find f Z ( z | x, y ). 5. (2 pts) (Textbook exercise 4.45) The number N of customer arrivals at a service station is a Poisson random variable with mean α customers per second. There are four types of customers. Let X k be the number of type k arrivals. Suppose P [ X 1 = k 1 , X 2 = k 2 , X 3 = k 3 | N = n ] = p ( k 1 , k 2 , k 3 ) = n ! p k 1 1 p k 2 2 p k 3 3 (1-p 1-p 2-p 3 ) n-k 1-k 2-k 3 k 1 ! k 2 ! k 3 !( n-k 1-k 2-k 3 )! a. Find the joint pmf of ( N, X 1 , X 2 , X 3 ). b. Find the marginal pmf of ( X 1 , X 2 , X 3 ). 1...
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