. . The Bernoulli distribution Independent trials having constant probability of success p were first studied by the Swiss mathematician Jacques Bernoulli ( – ). In his book The Art of Conjecturing , published by his nephew Nicholas eight years after his death in , Bernoulli showed that if the number of such trials were large, then the proportion of them that were successes would be close to p.
In probability theory, the Bernoulli distribution , named after Swiss mathematician Jacob Bernoulli is the discrete probability distribution of a random variable . A Bernoulli experiment is a random experiment, where the outcomes is classified in one of two mutually exclusive and exhaustive ways, say success or failure (example: heads or tails, defective item or good item, life or death or many other possible pairs). A sequence of Bernoulli trails occurs when a Bernoulli experiment is performed several independent times so that the probability of success remains the same from trial to trial. Any nontrivial experiment can be dichotomized to yield Bernoulli model.
Definition . : ( Bernoulli’s distribution) Let X be a random variable associated with a Bernoulli trial by defining it as X (success) = and X (failure) = , such that where then X is called a Bernoulli random variable and f x ( ) is called the Bernoulli distribution. Or equivalently If a random variable X is following a Bernoulli’s distribution, with probability p of success can be denoted as X Ber p ( ) , where p is called the parameter, then the probability mass function of X is Jacob Bernoulli ( - ) - - Probability Distributions The cumulative distribution of Bernoulli’s distribution is F x ( ) = = − ≥ if if if Mean : E X ) = x f x Note that, since X takes only the values and , its expected value p is “never seen”. Variance : V X ) = E X E X x f x = pq where q If X is a Bernoulli’s random variable which follows Bernoulli distribution with parameter p, the mean μ and variance σ are p and pq When p = , the Bernoulli’s distribution become for for and the cumulative distribution is F x ( ) = ≥ if if if The mean and variance are respectively are and