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The Bernoulli distribution is a discrete probability distribution with the only two possible values for the random variable. Each instance of an event with a Bernoulli distribution is called a Bernoulli trial.

The Bernoulli distribution uses the following parameter.

Parameter | Description | Support |
---|---|---|

`p` | Probability of success | $$0\le p\le 1$$ |

The probability mass function (pmf) is

$$f(x|p)=\{\begin{array}{c}1-p\text{\hspace{1em}},\text{\hspace{1em}}x=0,\\ \text{\hspace{1em}}p\text{\hspace{1em}},\text{\hspace{1em}}x=1\end{array}\text{\hspace{0.17em}}.$$

The mean is

$$\text{mean}=p\text{\hspace{0.17em}}.$$

The variance is

$$\mathrm{var}=p\left(1-p\right)\text{\hspace{0.17em}}.$$

The Bernoulli distribution is a special case of the binomial distribution,
with the number of trials *n* = 1. The geometric distribution models
the number of Bernoulli trials before the first success (or first
failure).

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