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.