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
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.