Binomial Distribution

We now develop the binomial probability distribution that is used extensively in many applied business and economic problems. Our approach begins by first developing the Bernoulli model, which is a building block for the Binomial. We consider a random experiment that can be give rise to just two possible mutually exclusive and collectively exhaustive outcomes,  which for convenience we will label “success” and “failure”. Let p denote the probability of success, so that the probability of failure is (1 – p). Now define the random variable X so that X takes the value 1 if the outcome of the experiment is success and 0 otherwise. The probability function of this random variable is then

                 P(X=0) = (1 – p)     and   P(X=1) = p

This distribution is known as Bernoulli distribution.

The Binomial Distribution

Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that P is the probability of a success in a single trial. If n independent trials are carried out, the distribution of the number of resulting success, x, is called the binomial distribution. Its probability distribution function for the binomial random variable X = x is