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13.7: Lotteries
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/13%3A_Games_of_Chance/13.07%3A_Lotteries
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 $\begin{align} \E(U) & = \frac{n^2}{N} \\ \var(U) & = \frac{n^2 (N - n)^2}{N^2 (N - 1)} \end{align}$

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