Distribution - Bernoulli

Gist

A simple probability distribution for anything with success or failure. Like flipping a coin.

Definiton

It's a discrete probability distribution with only two outcomes.

P (X = x) = {\begin{cases} p, & x = 1, \\ 1 - p, & x = 0, \end{cases}

Properties

Mean: $p$
Variance: $p (1 - p)$

Relationship

It's a special case of the Distribution - Binomial when there is only one event. For example, if $n$ (number of events) and $k$ (number of subset) are equal to 1:

P (X = 1) = (\binom{1}{1}) p^{1} (1 - p)^{1 - 1}

which then reduces down to:

P (X = 1) = p

Trivial.

The Distribution - Geometric models the number of i.i.d Bernoulli trials to get a success.