Home page for accesible maths 3.1 Discrete Probability Models 3.1.3 Binomial Random Variables:

\operatorname{\mathsf{Bin}}(n,\theta)

3.1.5 Poisson Random Variables:

\operatorname{\mathsf{Poisson}}(\lambda)

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3.1.4 Geometric Random Variables: $\operatorname{\mathsf{Geom}}(\theta)$

A model for the outcomes of experiments which count the number of $0$ values before the first $1$ in a sequence of independent Bernoulli trials (each with probability $\theta$ of a $1$ ). The sample space is $\{0,1,\ldots\}$ , and

\displaystyle p(r)=(1-\theta)^{r}\theta

for $r=0,1,\ldots$ , and $p(r)=0$ otherwise.

$\operatorname{\mathsf{E}}\left[{R}\right]=\frac{1-\theta}{\theta}$ ,
${\operatorname{\mathsf{Var}}}\left[{R}\right]=\frac{1-\theta}{\theta^{2}}$ .