Question

Let X and Y be two independent, nonnegative integer-valued random variables whose distribution has the property

P( X=x|X+Y=x+y)= binom m x binom n y binom m+n x+y

for all nonnegative integers x and y where m and n are given positive integers. Assume that P(X = 0) and P(Y = 0) are strictly positive. Show that both X and Y have binomial distributions with the same parameter p, the other parameters being m and n respectively.

Answer 1

We have

$\left\{ \begin{array}{c} \frac{P\left( X=x \right) P\left( Y=y \right)}{P\left( X+Y=x+y \right)}=\frac{C_{m}^{x}C_{n}^{y}}{C_{m+n}^{x+y}}\\ \frac{P\left( X=x+1 \right) P\left( Y=y-1 \right)}{P\left( X+Y=x+y \right)}=\frac{C_{m}^{x+1}C_{n}^{y-1}}{C_{m+n}^{x+y}}\\\end{array} \right. \Rightarrow \frac{P\left( X=x+1 \right) /P\left( X=x \right)}{P\left( Y=y \right) /P\left( Y=y-1 \right)}=\frac{\frac{m-x}{x+1}}{\frac{n-y+1}{y}}$

from which

$\frac{P\left( X=x+1 \right)}{P\left( X=x \right)}=\alpha \frac{m-x}{x+1}$

Then by induction

$P\left( X=x \right) =P\left( X=0 \right) \alpha ^xC_{m}^{x},x=0,1,...,m$

We have

$\sum_{x=0}^m{P\left( X=x \right)}=1\Rightarrow \sum_{x=0}^m{P\left( X=0 \right) \alpha ^xC_{m}^{x}}=1\Rightarrow \\\Rightarrow P\left( X=0 \right) \left( \alpha +1 \right) ^m=1\Rightarrow P\left( X=0 \right) =\frac{1}{\left( \alpha +1 \right) ^m}$

Denote $p=\frac{\alpha}{\alpha +1}\Rightarrow \alpha =\frac{p}{1-p}$

Then $P\left( X=x \right) =\frac{1}{\left( \alpha +1 \right) ^m}\alpha ^xC_{m}^{x}=C_{m}^{x}p^x\left( 1-p \right) ^{m-x}\Rightarrow X\sim Bin\left( m,p \right)$

Next,

$\frac{P\left( Y=y \right)}{P\left( Y=y-1 \right)}=\alpha \frac{n-y+1}{y}\Rightarrow \\\Rightarrow P\left( Y=y \right) =P\left( Y=0 \right) \alpha ^yC_{n}^{y}\\\sum_{y=0}^n{P\left( Y=y \right) =1}\Rightarrow P\left( Y=0 \right) \left( 1+\alpha \right) ^n=1\Rightarrow P\left( Y=0 \right) =\frac{1}{\left( \alpha +1 \right) ^n}\Rightarrow \\\Rightarrow P\left( Y=y \right) =\frac{1}{\left( \alpha +1 \right) ^n}\alpha ^yC_{n}^{y}=C_{n}^{y}p^y\left( 1-p \right) ^{n-y}\Rightarrow Y\sim Bin\left( n,p \right)$