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Binomial distribution


Binomial Distribution

Probability Mass FunctionThe 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
where
(n x) = n!/(x!(n-x)!
The following is the plot of the binomial probability density function for four values of p and n = 100.
plot of the binomial probability density function
Cumulative Distribution FunctionThe formula for the binomial cumulative probability function isF(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.
plot of the binomial cumulative distribution function
Percent Point FunctionThe 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.
plot of the binomial percent point function
Common Statistics
Meann*p
Modep*(n+1) - 1 <= x <= p*(n+1)
Range0 to N
Standard DeviationSQRT(n*p*(1-p))
Coefficient of VariationSQRT((1-p)/(n*p))
Skewness(1-2*p)/SQRT(n*p*(1-p))
Kurtosis3 - 6/n + 1/(n*p*(1-p))
CommentsThe binomial distribution is probably the most commonly used discrete distribution.
Parameter EstimationThe maximum likelihood estimator of p (n is fixed) isptilde = x/n
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the binomial distribution.

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