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Continuing our discussion of random variables, we consider the Binomial random variable which is a straight-forward extension of the Bernoulli random variable. The extension is that we perform the same Bernoulli random variable repeatedly. If each trial is independent, i.e. no trial influences any other, than the Binomial random variable is the total number of positive outcomes in the set of trials.

Here are some examples:

• number of heads in seven tosses of a coin (7,1/2)
• number of ones and twos in thirteen rolls of a six-sided die (13,1/3)
• number of A+ children out of three from the same couple whose blood types are both AO+ (3,3/4)

The notation we use for a binomial random variable is:

and we say `Y is a binomial random variable with a total of n trials and each trial has probability p.’

Compared with the notation for a Bernoulli, we have two parameters for this distribution. The numbers in parentheses of the above examples provide the n and p parameters respectively. The probability mass function is

This pmf looks almost exactly the same as the pmf for the Bernoulli random variable, the only noticeable difference is the coefficient in front. This term is known as the binomial coefficient and we say `n choose p’. This term provides the number of ways we can choose p objects from a total of n objects. It can be calculated using the following formula:

where the exclamation points are the factorial, i.e. n! = n x (n-1) x … x 2 x 1. The second difference between this pmf and that for the Bernoulli is that y can take on integers from 0 up to n. Bernoulli is actually just a special case of the binomial with n=1.

Bottom Line: A binomial random variable can be used to model the sum of independent Bernoulli random variables.