A10. If X and Y are independent binomial random variables with identical parameters n
and p, show analytically that the conditional probability of X, given that X + Y = m
is the hypergeometric distribution.
"X,Y\\sim Bin\\left( n,p \\right) \\Rightarrow X+Y\\sim Bin\\left( 2n,p \\right) \\\\P\\left( X=k|X+Y=m \\right) =\\frac{P\\left( X=k,Y=m-k \\right)}{P\\left( X+Y=m \\right)}=\\\\=\\frac{C_{n}^{k}p^k\\left( 1-p \\right) ^{n-k}C_{n}^{m-k}p^{m-k}\\left( 1-p \\right) ^{n-m+k}}{C_{2n}^{m}p^m\\left( 1-p \\right) ^{2n-m}}=\\\\=\\frac{C_{n}^{k}C_{n}^{m-k}}{C_{2n}^{m}}-hypergeometric\\,\\,with\\,\\,parameters\\,\\,2n,n,m"
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