Suppose we have a universe consisting of $m$ elements where $k$ of them are of type $1$ and $m-k$ are of the second type. You sample $n$ elements from the universe equally likely without looking. If $Z$ is the number of type $1$ elements in the sample, how can we express the expected value of $Z$?
My approach: $Z$ can be represented a sum of indicator random variables $I_1+I_2+...+I_n$. Thus, $\mathbb{E}(Z) = \mathbb{E}(I_1+I_2+...+I_n) = \mathbb{E}(I_1) + \mathbb{E}(I_2) + ...+\mathbb{E}(I_n)$.
Since any element in the universe is of type $1$ with probability $\frac{k}{m}$, $\mathbb{E}(Z) = \frac{k \cdot n}{m}$
Is my logic correct?
