From: israel@math.ubc.ca (Robert Israel) Subject: Re: Distribution of random variable? Date: 15 Mar 1999 23:08:46 GMT Newsgroups: sci.math,alt.sci.math.probability In article <7cji97$129$1@ffx2nh4.news.uu.net>, "Patrick Powers" writes: |> Generate random integers for 1 to 10 until their sum exceeds N. |> The random variable is the number of integers. What is the distribution of |> this random variable? The probability generating function of this random variable is g_N(z) = sum_{k=0}^infinity z^k P(X=k). These satisfy g_n(z) = 1 for n < 0, and g_n(z) = z/10 sum_{j=1}^10 g_{n-j}(z) for n >= 0. Let M be the 10 x 10 matrix [z/10 z/10 z/10 z/10 z/10 z/10 z/10 z/10 z/10 z/10 ] [ ] [1 0 0 0 0 0 0 0 0 0 ] [ ] [0 1 0 0 0 0 0 0 0 0 ] [ ] [0 0 1 0 0 0 0 0 0 0 ] [ ] [0 0 0 1 0 0 0 0 0 0 ] [ ] [0 0 0 0 1 0 0 0 0 0 ] [ ] [0 0 0 0 0 1 0 0 0 0 ] [ ] [0 0 0 0 0 0 1 0 0 0 ] [ ] [0 0 0 0 0 0 0 1 0 0 ] [ ] [0 0 0 0 0 0 0 0 1 0 ] and v0 = [1,1,1,1,1,1,1,1,1,1]^T. Then g_n(z) is the first component of M^(n-1) v0, i.e. the sum of the first row of M^(n-1). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2