I have the following HW question:
There are $N$ balls in a box, $m$ balls with an $S$ for success and $N-m$ balls with an $F$ for failure. Choose $n$ balls at random ($n\leq N$) and let $X$ = the number of successes.
a. Calculate the probability function $p(\cdot)=P(X=\cdot)$
b. Calculate $EX$
I have managed to solve the first part of the question: $P(X=k)=\frac{\binom{m}{k}\cdot\binom{N-m}{n-k}}{\binom{N}{n}}$
But I am having difficulty with the second part, I need to evaluate $EX=\sum_{k=0}^{\min\{n,m\}}\frac{\binom{m}{k}\cdot\binom{N-m}{n-k}}{\binom{N}{n}}\cdot k$
I have opened the binomials by definitions and I am stuck there.
How can I calculate $EX$ ? any help is appreciated!