416-1 - P when X is an exponential random variable with...

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Denker SPRING 2010 416 Stochastic Modeling - Assignment 1 Due Date: Monday, January 25, 2010 Problem 1: (Problem 51, page 92) A coin, having probability p for landing heads, is Fipped until head appears for the r -th time. Let N denote the number of Fips required. Calculate E [ N ]. Hint: Write N as a sum of geometric random variables. Problem 2: (Problem 56, page 93) There are n types of coupons. Each newly obtained coupon is, independently, type i with probability p i , i = 1 , ..., n . ±ind the expected number and the variance of the number of distinct types obtained in a collection of k coupons. Problem 3: (Problem 62, page 94) In deciding upon the appropriate premium to charge, insurance companies sometimes use the exponential principle, de²ned as follows. With X as the random amount that it will have to pay in claims, the premium charged by the insurance company is P = 1 a ln( E [ e aX ]) where a is some speci²c positive constant. ±ind
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Unformatted text preview: P when X is an exponential random variable with parameter λ , and a = αλ , where 0 < α < 1. Problem 4: (Problem 74, page 95) Let X 1 , X 2 , ... be a sequence of idependent, identically distributed continuous random variables. We say that a record occurs at time n if X n > max( X 1 , ..., X n-1 ). That is, X n is a record if it is larger than each of X 1 , ..., X n-1 . Show 1. P { a record occurs at time n } = 1 n . 2. E [number of records by time n ] = ∑ n i =1 1 i . 3. V ar (number of records by time n ) = ∑ n i =1 i-1 i 2 . 4. Let N = min { n : n > 1 and a record occurs at time n } . Show E [ N ] = ∞ . Problem 5: (Problem 76, page 96) Let X and Y be independent random variables with means μ X and μ Y and variances σ 2 X and σ 2 Y . Show that V ar ( XY ) = σ 2 X σ 2 Y + μ 2 Y σ 2 X + μ 2 X σ 2 Y ....
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This note was uploaded on 04/15/2010 for the course STAT 416 taught by Professor Denker during the Spring '08 term at Penn State.

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