eGeometricDistribution

Example There are $N_ {1}$ white balls and $N_{2}$ black balls in the jar. Let $X$

be the number of white balls when taking out $n $

balls one by one without undoing (non-restoration extraction). Find the probability that the number of white balls is $i$

Answer Note that

$\displaystyle P_{r}(X = i) = \frac{\binom{N_{1}}{i} \binom{N_{2}}{n-i}}{\binom{N}{n}}$

Random variable $X$ follows a hypergeometric distribution and we write

$\displaystyle X \sim H_{g}(N,N_{1},n)$

Also,

$\displaystyle E(X) = \frac{n N_{1}}{N}, \ V(X) = \frac{n N_{1}}{N} \cdot \frac{n - N_{1}}{N} \cdot \frac{N - n}{N - 1}$