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Discrete Distributions

Random variables with a Binomial distribution are generated from a process that follows the following five rules:

- The number of trials,
*n*, is known. - Each trial has two possible outcomes. We will refer to these as either “Success” and “Failure,” or a 1 and 0, respectively.
- The probability of success in each trial,
*p*, does not change from trial to trial. - The trials are independent; that is, the outcome of each trial is not influenced by the outcome of any other trial.
- The random variable is the number of successes in those
*n*trials.

In short, the Binomial random variable is the sum of *n* independent and identically distributed Bernoulli random variables.

The Binomial distribution has two parameters. The first is the number of trials, *n*. The second parameter is the success probability, *p*. This latter parameter may also be represented as π, depending on the source. Note that the sample space is defined as all integers between 0 and *n*, inclusive.

Please select the aspect of the Binomial distribution you would like to work with:

Here is a graphic of a sample Binomial distribution:

In this example,

X ~ *Bin*(n=7, p=0.67)

The sample space is listed across the bottom, S = {0, 1, 2, … 7}. The height of each bar represents the probability of that elementary event.

© Ole J. Forsberg, Ph.D. 2018. All rights reserved. | . | |