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$X$ ~ $B(n,p)$
$X$ = 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
$p+q=1\phantom{\rule{15pt}{0ex}}q=1-p$
The mean is $\mu =\mathrm{np}$ . The standard deviation is $\sigma =\sqrt{\mathrm{npq}}$ .
Formula$X$ ~ $G\left(p\right)$
$X$ = 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 $\sigma =\sqrt{\frac{1}{p}(\left(\frac{1}{p}\right)-1)}$
Formula$X$ ~ $H(r,b,n)$
$X$ = 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: $\sigma =\sqrt{\frac{rbn(r+b-n)}{{(r+b)}^{2}(r+b-1)}}$
Formula$X$ ~ $\text{P(\mu )}$
$X$ = 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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