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I think the summation in $E(X)$ should be from $j$ to $N$, rather than $j$ to $n$.
Also, a simpler solution:
For each person, there's a $\frac{m}{N}$ chance that the first research picks them, and a $\frac{n}{N}$ chance that the second researcher picks them. So, there's a $\frac{mn}{N^2}$ chance for each person to be picked. By linearity, the expected overlap should be $\frac{mn}{N}$. Please correct me if I'm wrong.
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I think the summation in$E(X)$ should be from $j$ to $N$ , rather than $j$ to $n$ .
Also, a simpler solution:
For each person, there's a$\frac{m}{N}$ chance that the first research picks them, and a $\frac{n}{N}$ chance that the second researcher picks them. So, there's a $\frac{mn}{N^2}$ chance for each person to be picked. By linearity, the expected overlap should be $\frac{mn}{N}$ . Please correct me if I'm wrong.
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