4.8 Summary of functions

Binomial

$X$ ~ $B(n,p)$

$X$ = 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

$p+q=1q=1-p$

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

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 $\sigma =\sqrt{\frac{1}{p}(\left(\frac{1}{p}\right)-1)}$

Hypergeometric

$X$ ~ $H(r,b,n)$

$X$ = 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.

$n\le r+b$

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

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

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 $\sigma =\sqrt{\mu }$