Binomial distribution

Binomial Distribution

Probability Mass Function

The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled "success" and "failure". The binomial distribution is used to obtain the probability of observing x successes in N trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials.The formula for the binomial probability mass function is
P(x,p,n) = (n x)*(p**x)*(1-p)**(n-x) for x = 0, 1, 2, ..., n

P(x,p,n) = (n x)*(p**x)*(1-p)**(n-x) for x = 0, 1, 2, ..., n

where

The following is the plot of the binomial probability density function for four values of p and n = 100.

Cumulative Distribution Function

The formula for the binomial cumulative probability function is F(x,p,n) = SUM[(n i)*(p**i)*(1-p)**(n-i)
where the summation is over i from 0 to x

The following is the plot of the binomial cumulative distribution function with the same values of p as the pdf plots above.

Percent Point Function

The binomial percent point function does not exist in simple closed form. It is computed numerically. Note that because this is a discrete distribution that is only defined for integer values of x, the percent point function is not smooth in the way the percent point function typically is for a continuous distribution.The following is the plot of the binomial percent point function with the same values of p as the pdf plots above.