computing expectations for probability distributions in mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg105288] computing expectations for probability distributions in mathematica*From*: per <perfreem at gmail.com>*Date*: Wed, 25 Nov 2009 23:04:31 -0500 (EST)

hi all, i am trying to see if there is a simplified expression for a particular random variable using mathematica. i have two independent binomial random variables, X and Y, and i'd like to compute the expectation of X/(X+Y). more formally, X is distributed Binomial(p1, n) and Y is distributed Binomial(p2, n). note that p1 and p2 can be different, but the n's are the same. i would like to compute E(X/(X +Y)). since the ratio is undefined when X+Y = 0, i would like to assume that X+Y >= 1. is there a way to compute this expectation and see if it has a simplified analytic form subject to this constraint? i defined my function to compute the probability of this random variable as follows: myvar[x_, y_] := Binomial[n, x]*p1^x*(1 - p1)^(n - x)*Binomial[n, y]* p2^y*(1 - p2)^(n - y)*(x/(x + y)) now i just want to take the expectation of this. since the value of X ranges from 0 to N and the value of Y ranges from 0 to N, it should be possible to simply take a sum over X from 0 to N and sum of Y from 0 to Y to see what this expression evaluates to and whether it has a simple analytic form. the trick is to take the sum with the constraint of X + Y >= 1 -- how can i express this in Mathematica? i am also open to computing the expectation of X/(X+Y) by assuming they are independent Poissons if that makes the math easier, i.e. using poisson approximation to binomial. thank you for your help. > > where X, Y are independent but p1 and p2 could be different. I am > > trying to compute the mean E(X/(X+Y)) but i am stuck. is there an easy > > form for this mean? i looked it up in several probability books but > > could not get it. > > > i am willing to make either a Poisson approximation or a Normal > > approximation to thebinomialfor my application if i could compute > > this mean. if i assume that X and Y are distributed Poisson, i know > > that E(X+Y) has a nice form as a function of the rates of the two > > poissons, but i still do not know how to get the distribution of X/(X > > +Y) so that i can compute the expectation E(X/(X+Y)). >

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