I am studying my probability course and came up with the following question:
Let $X, Y, Z$ be three independent geometrically distributed random variables with parameter $p, q, l$ which represents the number of failures before the first success. In addition, those variables are conditioned by $X+Y+Z=A$. Let $n,m,k \in N$ and such that $n+m+k=B$. Find $$ E(X^n\times Y^m\times Z^k|X+Y+Z=A) $$
I have tried to represent $Z=A-X-Y$ and use nested expectations, but it did not give any results.
I am considering a probability first:
$$ P(X^n\times Y^m\times Z^k|X+Y+Z=A)=\frac{P(X^n\times Y^m\times Z^k)P(X=A-Y-Z)}{P(X+Y+Z=A)} $$ The denominator can be easily computed, however, I do not know how to compute nominator.