Geometric and Hypergeometric Distributions

Geometric Distribution

Geometric Distribution is the probability distribution of the number $X$ of Bernoulli trails needed to get one success, supported on $X=1,2,\dots$

The PMF of geometric distribution is

$$P(X=k)=(1-p{)}^{k-1}p$$

and $\mathrm{E}[X]=\frac{1}{p}$

Hypergeometric Distribution

The hypergeometric distribution describes the probability of $k$ success in $n$ draws, without replacement, from a finite population of size $N$ that contains exactly $K$ objects with that feature, wherein each draw is either a success or a failure.

Tip

In contrast, the binomial distribution describe the probability of $k$ successes in $n$ draws with replacement

The PMF of hypergeometric distribution is

$$P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{N}{k}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$$

and $\mathrm{E}[X]=n\cdot \frac{K}{N}$