416-1sol - Denker SPRING 2010 416 Stochastic Modeling -...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Denker SPRING 2010 416 Stochastic Modeling - Assignment 1 SOLUTIONS Problem 1: (Problem 51, page 92) A coin, having probability p for landing heads, is flipped until head appears for the r-th time. Let N denote the number of flips required. Calculate E [ N ]. Hint: Write N as a sum of geometric random variables. Solution: Let X j denote the waiting time for the j-th head to show up. We claim that X j is a geometric random variable with parameter p . Indeed, if head appears for the j 1-th time, then flipping the coin is just like starting to flip a coin, so the probability of appearing head first in the n-th trial is the same as the probability of having a head the first time in the n-th trial after the j 1-th appearance of head. So, N = r i =1 X i is the waiting time for the r-th head to show up and EN = r summationdisplay i =1 EX i = rEX 1 = r p is the expected time (see book page 68/69 for the expectation of a geometric random variable with parameter p ). 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 . Find the expected num-....
View Full Document

This note was uploaded on 04/15/2010 for the course STAT STAT 416 taught by Professor Prof.denker during the Spring '10 term at Peru State.

Page1 / 3

416-1sol - Denker SPRING 2010 416 Stochastic Modeling -...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online