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Hw01 - STAT 410 Homework#1(due Friday January 25 by 3:00...

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STAT 410 Spring 2008 Homework #1 (due Friday, January 25, by 3:00 p.m.) 1. Consider a continuous random variable X with probability density function f X ( x ) = ° ± ° ² ³ < < o.w. 0 1 0 3 2 x x Find the moment-generating function of X, M X ( t ) . 2. Suppose a discrete random variable X has the following probability distribution: P( X = k ) = ( ) ! 2 ln k k , k = 1, 2, 3, … . a) Verify that this is a valid probability distribution. b) Find μ X = E ( X ) by finding the sum of the infinite series. c) Find the moment-generating function of X , M X ( t ) . d) Use M X ( t ) to find μ X = E ( X ) . 3. Suppose a random variable X has the following probability density function: ° ± ° ² ³ = - otherwise 0 1 0 ) ( x C x f x e a) What must the value of C be so that f ( x ) is a probability density function? b) Find the cumulative distribution function F ( x ) = P ( X x ) . c) Find the median of the probability distribution of X . d) Find μ X = E ( X ) .
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4. Suppose a random variable X has the following probability density function: ° ± ° ² ³ = otherwise 0 1 1 ) ( C x x x f a) What must the value of
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