Let nj t be the number of i n t with yi j in example

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: · · + YN (t) where we set S (t) = 0 if N (t) = 0. In Example 2.1, S (t) gives the number of customers that have arrived up to time t. In Example 2.2, S (t) represents the total number of bytes in all of the messages up to time t. In each case it is interesting to know the mean and variance of S (t). Theorem 2.10. Let Y1 , Y2 , . . . be independent and identically distributed, let N be an independent nonnegative integer valued random variable, and let S = Y1 + · · · + YN with S = 0 when N = 0. (i) If E |Yi |, EN < 1, then ES = EN · EYi . (ii) If EYi2 , EN 2 < 1, then var (S ) = EN var (Yi ) + var (N )(EYi )2 . (iii) If N is Poisson( ), then var (S ) = E Yi2 . 87 2.3. COMPOUND POISSON PROCESSES Why is this reasonable? The first of these is natural since if N = n is nonrandom ES = nEYi . (i) then results by setting n = EN . The formula in (ii) is more complicated but it clearly has two of the necessary properties: If N = n is nonrandom, var (S ) = n var (Yi ). If Yi = c is nonrandom var (S ) = c2 var (N ). Combining...
View Full Document

This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

Ask a homework question - tutors are online