In this case, the probability density function of the number of catches \(U\) is \[ \P(U = k) = \frac{\binom{n}{k} \binom{N - n}{n - k}}{\binom{N}{n}}, \quad k \in \{0, 1, \ldots, n\} \] The mean and ...In this case, the probability density function of the number of catches \(U\) is \[ \P(U = k) = \frac{\binom{n}{k} \binom{N - n}{n - k}}{\binom{N}{n}}, \quad k \in \{0, 1, \ldots, n\} \] The mean and variance of the number of catches \(U\) in this special case are \begin{align} \E(U) & = \frac{n^2}{N} \\ \var(U) & = \frac{n^2 (N - n)^2}{N^2 (N - 1)} \end{align}
