Binomial Distribution

Consider a statistical experiment where a success occurs with probability p and a failure occurs with probability q = 1 - p. If this experiment is repeated for n times, with each repetition or trial independent of the other, then the random variable X, whose value is the number of successes in the n trials, has a binomial distribution with parameters p and n. The random variable X is said to be binomially distributed with parameters p and n, and has the following probability mass function:

The binomial term epresents the number of ways that x successes can be

distributed among n trials, while p^x(1 - p)^n-x is the probability of having n successes and (n - x) failures. Thus, the product of these two terms is the probability of having x successes in the n trials.

A random variable X that is binomially distributed with parameters p and n is the sum of n independent Bernoulli variables with parameter p. Thus, the mean and variance of a binomially distributed random variable X are calculated as follows:

EX. A regular coin, when tossed, will come up heads or tails with equal probability. The random variable H, whose value is the number of heads in n trials, is binomially distributed with parameters n and p = 0.5. Its probability mass function has the following form:

For n = 4, the probability mass function is