16.1: Binomial Distribution

The Binomial distribution is defined as the sum of independent and identically distributed Bernoulli random variables. Subsequently, this results in five requirements for a random variable to follow a Binomial distribution:

1. the number of trials, \(n\), is known;
2. each trial has two possible outcomes: success and failure;
3. the success probability for each trial, \(\pi\), is constant;
4. each trial is independent from the others; and
5. the random variable is the number of successes in those \(n\) trials.

One can also think of the Binomial distribution as a generalization of the Bernoulli distribution to \(n > 1\). Similarly, one can think of the Bernoulli distribution as a special case of the Binomial, where \(n=1\). Whichever way you look at it, there are a number of similarities between the two distributions.

The sample space for the Binomial distribution is key to understanding when it can be used, \(\mathcal{S} = \{0, 1, 2, \ldots, n\}\). Thus, the Binomial distribution can be used to model counts of successes when the number of attempts (trials) is known. The following are variables that could follow a Binomial distribution:

• number of students passing a class
• number of Euchre games a person wins in a tournament finals
• number of college students in a class who can locate Ruritania on a map
• number of football games the SUR Hawks win in a year
• number of pages in a book that have a typographical error
• number of fireworks in a shipment of 144 that are duds
• number of cast ballots declared invalid in an electoral division

Note that each of these examples start with "number of." This is because the Binomial distribution models the "number of" successes (out of a given number of tries). Second, note that the sample space has both a lower (no successes) and an upper bound (no failures). The upper bound for the first example is the number of students taking that statistics class. The upper bound for the second is the number of hands played by a person in a poker game; for the third, the number of students in that class; for the fourth, the number of games the SUR Hawks play in a year; etc.

The following variables cannot follow a Binomial distribution:

• number of crimes in Děčín this year
• number of injuries experienced by the SUR Hawks
• number of errors in a book
• number of dents on a car

While each of these also starts with "the number of," none of these can follow a Binomial distribution. In each case, there is no upper bound. The first example measures the number of successes over a time period. There can be multiple crimes on a given day, so there is no upper bound. To make this a Binomial random variable, one could measure instead the number of Děčín residents who are the target of a crime in a given year. In that case, there is an upper bound — the population of Děčín.

The second example also has no upper bound. Each player can have multiple injuries. To make this a possible Binomial random variable, one could measure the number of players injured. Note that the upper bound would then be set at the number of players on the Hawks.

It is interesting that these four random variables could be examples of Poisson random variables. We will be covering how to analyze such count data later (see the chapter on count models).

Note

One thing that may help you determine whether a variable follows a Binomial or a Poisson distribution could be this: If you can represent a similar variable as a proportion, then it is Binomial; if as a rate, then Poisson.