Bernoulli distribution

Bernoulli Distribution is a discrete probability distribution for a random variable which can take a binary outcome: success (usually denoted by 1) or failure (usually denoted by 0). Named after Swiss mathematician Jacob Bernoulli, it is a special case of the two-point distribution, for which the outcome need not be a bit, and the two outcomes need not be equally likely.

Definition

A random variable X is said to have a Bernoulli distribution if it takes two values, 1 and 0, with probabilities p and 1-p respectively, where 0 < p < 1. This can be succinctly expressed using the probability mass function (PMF):

f(k) = p^k(1-p)^1-k for k ∈ {0,1}

Properties

The Bernoulli distribution has several important properties:

• Expected value: The expected value of a Bernoulli random variable X is p.
• Variance: The variance of a Bernoulli random variable X is p(1-p).
• Skewness: The skewness of a Bernoulli random variable X is (1-2p)/√(pq), where q=1-p.
• Kurtosis: The kurtosis of a Bernoulli random variable X is 1-6pq.

Applications

The Bernoulli distribution is widely used in various fields such as statistics, economics, psychology, and computer science. It is particularly useful in scenarios where outcomes are binary, such as coin tosses, success/failure experiments, and yes/no surveys.

Related Distributions

The Bernoulli distribution is related to several other probability distributions:

• The sum of n independent Bernoulli trials follows a binomial distribution.
• The number of trials needed to get the first success in a series of Bernoulli trials follows a geometric distribution.
• The time until the first success in a series of Bernoulli trials follows an exponential distribution.

See Also

• Binomial distribution
• Geometric distribution
• Exponential distribution
• Probability theory
• Statistics