📂Probability Distribution

Bernoulli Distribution

Definition¹

For $p \in [0, 1]$, a discrete probability distribution with the following probability mass function is referred to as a Bernoulli distribution.

$$ f(x) = p^{x}(1-p)^{1-x}, \qquad x = 0, 1 $$

Description

This distribution is used when describing an experiment with only two possible outcomes, such as a coin toss. Because there are two possible outcomes, $x = 1$ is commonly referred to as a success and $x = 0$ as a failure. The probability of success is denoted as $p$, and the probability of failure as $q = 1 - p$. Conducting an experiment where there are only two possible outcomes is known as a Bernoulli trial.

When the number of trials is generalized to $n$ times, it results in a binomial distribution. Conversely, a Bernoulli distribution can be viewed as a special case of the binomial distribution when $n = 1$ becomes $\operatorname{Bin}(1, p)$.

When the possible outcomes (categories) are generalized from two to $k$, it becomes a categorical distribution. If both the number of trials and categories are generalized, it becomes a multinomial distribution.

Category Number of Trials	$1$ times	$n$ times
$2$ categories	Bernoulli distribution	Binomial distribution
$k$ categories	Categorical distribution	Multinomial distribution

Basic Properties

Moment Generating Function

The moment generating function of a Bernoulli distribution is as follows.

$$ m(t) = 1 - p + pe^{t} = q + pe^{t}, \qquad q = 1 - p $$

Mean and Variance

If $X \sim \operatorname{Bin}(1, p)$, then

$$ E(X) = p $$ $$ \Var(X) = p(1-p) = pq, \qquad q = 1 - p $$