# 4.6 Summary of functions

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This module provides a review of the binomial, geometric, hypergeometric, and Poisson probability distribution functions and their properties.
Formula

## Binomial

$X$ ~ $B\left(n,p\right)$

$X$ = 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

$p+q=1\phantom{\rule{15pt}{0ex}}q=1-p$

The mean is $\mu =\mathrm{np}$ . The standard deviation is $\sigma =\sqrt{\mathrm{npq}}$ .

Formula

## Geometric

$X$ ~ $G\left(p\right)$

$X$ = the number of independent trials until the first success (count the failures and 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 $\mu =\frac{1}{p}$

Τhe standard deviation is $\sigma =\sqrt{\frac{1}{p}\left(\left(\frac{1}{p}\right)-1\right)}$

Formula

## Hypergeometric

$X$ ~ $H\left(r,b,n\right)$

$X$ = the number of items from the group of interest that are in the chosen sample.

$X$ may take on the values $x$ = 0, 1, ..., up to the size of the group of interest. (The minimum valuefor $X$ may be larger than 0 in some instances.)

$r$ = the size of the group of interest (first group)

$b$ = the size of the second group

$n$ = the size of the chosen sample.

$(n, r+b)$

The mean is: $\mu =\frac{nr}{r+b}$

The standard deviation is: $(\sigma , \sqrt{\frac{rbn\left(r+b-n\right)}{{\left(r+b\right)}^{2}\left(r+b-1\right)}})$

Formula

## Poisson

$X$ ~ $\text{P(μ)}$

$X$ = the number of occurrences in the interval of interest

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

The mean $\mu$ is typically given. ( $\lambda$ is often used as the mean instead of $\mu$ .) When the Poisson is used to approximate the binomial, we use the binomialmean $\mu =np$ . $n$ is the binomial number of trials. $p$ = the probability of a success for each trial. This formula is valid when n is "large" and $p$ "small" (a general rule is that $n$ should be greater than or equal to $20$ and $p$ should be less than or equal to $0.05$ ). If $n$ is large enough and $p$ is small enough then the Poisson approximates the binomial very well. The variance is ${\sigma }^{2}=\mu$ and the standard deviation is $\sigma =\sqrt{\mu }$

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