From: israel@math.ubc.ca (Robert Israel)
Subject: Re: A probability problem!
Date: 28 Jan 2000 19:20:38 GMT
Newsgroups: sci.math
Summary: [missing]

In article <86r4ka$brn$1@eng-ser1.erg.cuhk.edu.hk>,
 "Ding Xiaowei" <xwding9@ie.cuhk.edu.hk> writes:

> I have S as a sack-bag,
> 
> Then I have several items to put into this bag, the sizes of items have a
> kind of distribution, i.e. Xi, i=1,2,... have a distribution,
> 
> Then I want to put these items into this sack-bag, satisfying the following
> conditions:
> 
> X1+X2+...Xk<=S
> X1+X2+...X(k+1)>S
> 
> Then I want to compute the expectation of X1+X2+...Xk, how can I do?

This is a problem in Renewal Theory.  Suppose the X_i have a density f(x), 
with mean mu and variance v.
Let S_j = X1 + ... + Xj, and N(t) = max{j: S_j <= t).  What you want
is then S_{N(S)}.  Its expected value g(t) = E[S_{N(t)}] satisfies an 
integral equation

g(t) = int_0^t (x + g(t-x)) f(x) dx

If I'm not mistaken, g(t) = t - (v + mu^2)/(2 mu) + o(t) as t -> infinity. 

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2