## 4.1 Hypergeometric Distribution

$h\left(x\right)=\frac{\left(\begin{array}{c}A\\ x\end{array}\right)\left(\begin{array}{c}N-A\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$

## 4.2 Binomial Distribution

*X* ~ *B*(*n*, *p*) means that the discrete random variable *X* has a binomial probability distribution with *n* trials and probability of success *p*.

*X* = the number of successes in *n* independent trials

*n* = the number of independent trials

*X* takes on the values *x* = 0, 1, 2, 3, ..., *n*

*p* = the probability of a success for any trial

*q* = the probability of a failure for any trial

*p* + *q* = 1

*q* = 1 – *p*

The mean of *X* is *μ* = *np*. The standard deviation of *X* is *σ* = $\sqrt{npq}$.

where P(X) is the probability of X successes in n trials when the probability of a success in ANY ONE TRIAL is p.

## 4.3 Geometric Distribution

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

*X* ~ G(*p*) means that the discrete random variable *X* has a geometric probability distribution with probability of success in a single trial *p*.

*X* = the number of independent trials until the first success

*X* takes on the values *x* = 1, 2, 3, ...

*p* = the probability of a success for any trial

*q* = the probability of a failure for any trial *p* + *q* = 1 *q* = 1 – *p*

The mean is *μ* = $\frac{1}{p}$.

The standard deviation is *σ* = $\sqrt{\frac{1\text{}\u2013\text{}p}{{p}^{2}}}$
= $\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}$
.

## 4.4 Poisson Distribution

*X* ~ *P*(*μ*) means that *X* has a Poisson probability distribution where *X* = the number of occurrences in the interval of interest.

*X* takes on the values *x* = 0, 1, 2, 3, ...

The mean *μ* or *λ* is typically given.

The variance is *σ*^{2} = *μ*, and the standard deviation is

$\sigma \text{=}\sqrt{\mu}$.

The probability of having exactly $x$ successes in $r$ trials is $P\left(X=x\right)=\left({e}^{-\mu}\right)\frac{{\mu}^{x}}{x!}$.

When *P*(*μ*) is used to approximate a binomial distribution, *μ* = *np* where *n* represents the number of independent trials and *p* represents the probability of success in a single trial.