Hypergeometric Distribution

def. Hypergeometric Distribution. Describes probabilities with successive draws without replacement. In a population of $N$ with $K$ members containing the feature, among $n$ trials the probability of drawing $k$ with the feature is:

$$\begin{array}{rl}X& \sim Hypergeometric[N,K,n]\\ \mathbb{P}(X=k)& =\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{K_k(N-K{)}_{n-k}}{N_n}\\ & =\frac{\left(\genfrac{}{}{0ex}{}{K}{k}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}\end{array}$$

def. Geometric Distribution. let $X$ s.t. $\text{Range}(X)=\mathbb{N}$, where $X$ is the number of trials until the first success, and success probability is $p$. $X$ is Geometrically p-Distributed:

$$\begin{gathered} X\sim \text{Geom}(p) \\ \mathbb{P}(X=k)=p(1-p{)}^{k-1} \end{gathered}$$

• where $X$ is the number of total trials and $p$ is the probability of an event—normally, geometric distributions are used to model the number of successful events before a failure.
• e.g. the number of basketball free throws until a failure
• $\mathbb{E}(X)=\frac{1}{p}$
• $\mathbb{SD}(X)=\frac{\sqrt{1-p}}{p}$

Info

$Y=X-1$ is also a geometric distribution; the number of failures before a success.

$$\begin{gathered} >Y\sim \text{Geom}(p) \\ >\mathbb{P}(Y=k)=p(1-p{)}^k> \end{gathered}$$

def. Negative Binomial Distribution. let $X$ describe the number of successes with probability $p$ before $r$ failures. $X$ is a Negative Binomal Distribution where:

$$\begin{gathered} X\sim \text{NegBinom}(r,p) \\ \mathbb{P}(X=k)=\left(\genfrac{}{}{0ex}{}{k+r-1}{k}\right)(1-p{)}^r{p}^k \end{gathered}$$

• $\mathbb{E}(X)=\frac{pr}{1-p}$
• $\mathbb{SD}(X)=\frac{\sqrt{pr}}{1-p}$