# 410Hw01 - STAT 410 Summer 2009 Homework#1(due Friday June...

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STAT 410 Summer 2009 Homework #1 (due Friday, June 19, by 4:00 p.m.) 1. Suppose a discrete random variable X has the following probability distribution: P( X = k ) = ( 29 ! 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 ). e) Find σ X 2 = Var ( X ). 2. Suppose a random variable X has the following probability density function: = otherwise 0 1 1 ) ( C x x x f a) What must the value of C be so that f ( x ) is a probability density function? b) Find P ( X < 2 ). c) Find P ( X < 3 ). d) Find μ X = E ( X ). e) Find σ X 2 = Var ( X ).

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3. An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, Y, follows a distribution with density function: f ( y ) = otherwise 0 1 if 2 3 y y a) What is the expected value and the variance of the policyholder’s loss? b)
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## This note was uploaded on 10/29/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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410Hw01 - STAT 410 Summer 2009 Homework#1(due Friday June...

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