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Binomial Distribution

A binomial random variable is the number of successes in a series of trials, for example, the number of ‘heads’ occurring when a coin is tossed 50 times.

A discrete random variable X is said to follow a Binomial distribution with parameters n and p if it has probability distribution

P(X=x) = (n choose x).p^x.(1-p)^n-x


x = 0, 1, 2, … , n

n = 1, 2, 3, …

p = success probability; 0 < p < 1

(n choose x) = n! / {x!(n-x)!}

The trials must have the following characteristics:

  1. the total number of trials is fixed in advance;
  2. there are just two possible outcomes of each trial: success or failure;
  3. the outcomes of all the trials are statistically independent;
  4. all the trials have the same probability of success.


Mean (µ) of the Binomial Distribution:

µ = np

Variance (σ2) of the Binomial Distribution:

σ2 = np(1-p)


Easton, V. J., McColl, J.H. (September 1997). Statistics Glossary – random variables and probability distributions. Retrieved December 2, 2006 from http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#binodistn.



1. Lisa - October 20, 2011

easy to understand! good!

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