Home page for accesible maths 3.1 Discrete Probability Models 3.1.2 Bernoulli Random Variables:

\operatorname{\mathsf{Bern}}(\theta)

3.1.4 Geometric Random Variables:

\operatorname{\mathsf{Geom}}(\theta)

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3.1.3 Binomial Random Variables: $\operatorname{\mathsf{Bin}}(n,\theta)$

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

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

for $r=0,1,2,\dots,n$ , and $p(r)=0$ otherwise.

$\operatorname{\mathsf{E}}\left[{R}\right]={\color[rgb]{0.76,0.01,0}n\theta,}$
${\operatorname{\mathsf{Var}}}\left[{R}\right]={\color[rgb]{0.76,0.01,0}n\theta% (1-\theta).}$