Re: computing expectations for probability distributions in mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg105297] Re: computing expectations for probability distributions in mathematica*From*: dh <dh at metrohm.com>*Date*: Thu, 26 Nov 2009 06:15:29 -0500 (EST)*References*: <hekuoc$9an$1@smc.vnet.net>

per wrote: > 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)). > Hi, you can split your sum into 3 sums: x=0; y=1..n y=0; x=1..n; x=1..n; y=1..n However, are you sire that you can simply eliminate the case x=0 and y=0? Otherwise the expectation value does not exist. Daniel