Bernoulli Distribution:

A Bernoulli trial is one of the simplest experiments with exactly two possible outcomes, success and failure.

Example:

• Coin tosses: Number of heads up/number of tails up.
• Births: number of boys/girls born each day.
• Rolling Dice: the probability of two die roll resulting in a double six.

A Bernoulli distribution is a discrete probability distribution in which the random variable (X) takes only two possible values (Bernoulli trial). One possible value is 1 (success) with probability p and another value is 0 (failure) with probability (1–p). Here, p denotes the probability of success.

The Bernoulli probability distribution of the random variable X is given by,

It can also be rewritten as,

The expected value (MEAN) of a Bernoulli distribution is

E(X) = 0×(1-p) + 1×p = p.

The variance of Bernoulli distribution is

Var (X) = E(X²) – E(X)² = [1² ×p + 0² × (1-p) – p² ] = p – p² = p(1-p).

The mode of a Bernoulli distribution (the value with the highest probability of occurring) is

Mode =  

The Bernoulli distribution can also be defined as the Binomial distribution with n = 1.

The Bernoulli distribution is sometimes used to model a single individual experiencing an event like death, a disease, or disease exposure in clinical trials. The occurrence of a disease can be modeled in logistic regression with help of Bernoulli distributions.