If X and Y are independent binomial random variables with id

If X and Y are independent binomial random variables with identical parameters n and
p, calculate the conditional expected value of X given X+Y = m.
Hint: Begin by calculating the conditional mass function of X given that X+Y = m.
Use the fact that X+Y is a binomial distribution with parameters 2n and p.

Solution

X and y are binomial independent random variables.

Both have parameters as p and y

Hence x+y will have parameter as p and 2n.

E(x/x+y) = E(x intersection x+y)/E(x+y)

BUt x and y are independent

Hence x and x+y are independent

Mass function of x given x+y will be the same as mass function of X

2n = m

Thus pdf of X = nCr p^rq^n-r