Bernoulli Distribution

Definition:

The Bernoulli distribution is a discrete probability distribution that models the outcomes of a single binary experiment. It represents a random variable that takes on two possible outcomes, usually labeled as success or failure, with corresponding probabilities of success, p, and failure, q = 1 – p, where 0 <= p <= 1.

Formal Notation:

X ~ Bernoulli(p)

Parameters:

  • p – The probability of success for the given binary experiment.

Probability Mass Function (PMF):

The probability mass function of the Bernoulli distribution is defined as:

P(X = k) = pk * (1 – p)1 – k

where k is the outcome of the binary experiment, with k = 0 indicating failure and k = 1 indicating success.

Mean and Variance:

The mean (or expected value) of a Bernoulli random variable is given by:

E(X) = p

The variance of a Bernoulli random variable is given by:

Var(X) = p(1 – p)

Use Cases:

  • The Bernoulli distribution is commonly used in scenarios where there are only two possible outcomes, such as tossing a coin (heads or tails), passing or failing an exam, or a customer making a purchase (yes or no).
  • It serves as the building block for more complex probability distributions, such as the binomial distribution.
  • It is often used in statistical modeling and hypothesis testing.