410Pr1 - STAT 410 Practice Problems Fall 2009 1 Suppose a...

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STAT 410 Practice Problems Fall 2009 1. Suppose a random variable X has the following probability density function: f ( x ) = 1 x / 2 0 x 2 f ( x ) = 0 otherwise a) Find the cumulative distribution function F (x) = P(X x). b) Find the median of the probability distribution of X. c) Find the probability P( 0.8 X 1.8 ). d) Find μ X = E ( X ). e) Find σ X 2 = Var ( X ). f) Find the moment-generating function of X. 2. 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 ( 29 ( 29 + = otherwise 0 1 0 , 1 0 , 2 y x y x k y x f a) Find the value of k that would make ( 29 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
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410Pr1 - STAT 410 Practice Problems Fall 2009 1 Suppose a...

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