Practice Problems 5 - Practice Problems 3 1. A bank...

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Unformatted text preview: Practice Problems 3 1. A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of time that the walk-up window is in use. Then the set of possible values for ( X , Y ) is the rectangle D = { ( x , y ) : 0 x 1, 0 y 1 } . Suppose the joint probability density function of ( X , Y ) is given by ( ) ( ) + = otherwise 1 , 1 , 2 y x y x k y x f a) Find the value of k that would make ( ) y x f , a valid probability density function. b) Find the probability that neither facility is busy more than one-quarter of the time. That is, find P ( 0 X , 0 Y ). c) What are the marginal probability density functions of X and Y? Are X and Y independent? d) Find P ( Y ). 2. Let X and Y be two independent random variables, with probability density functions f X ( x ) and f Y ( y ) , respectively. f X ( x ) = x e x- , x > 0, f Y ( y ) = y e- , y > 0. Find the probability P ( X < Y ). 3. Let X and Y have the joint probability density function f X Y ( x , y ) = x , x > 0, 0 < y < e x , zero elsewhere. a) Find f X ( x ) and f Y ( y ). b) Find E ( X ) and E ( Y ). c) Are X and Y independent? 4. Let X 1 denote the number of customers in line at the express checkout and X 2 denote the number of customers in line at the regular checkout at a local market. Suppose the joint probability mass function of X 1 and X 2 is as given in the table below. x 2 0 1 2 3 x 1 0 0.08 0.07 0.04 0.00 1 0.06 0.15 0.05 0.04 2 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 4 0.00 0.01 0.05 0.06 a) Find P ( X 1 = X 2 ), that is, find the probability that the number of customers...
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Practice Problems 5 - Practice Problems 3 1. A bank...

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