Binomial vs. Poisson Distribution: Similarities & Differences


Two distributions that are similar in statistics are the Binomial distribution and the Poisson distribution.

This tutorial provides a brief explanation of each distribution along with the similarities and differences between the two.

The Binomial Distribution

The Binomial distribution describes the probability of obtaining k successes in n binomial experiments.

If a random variable X follows a binomial distribution, then the probability that X = k successes can be found by the following formula:

P(X=k) = nCk * pk * (1-p)n-k

where:

  • n: number of trials
  • k: number of successes
  • p: probability of success on a given trial
  • nCkthe number of ways to obtain k successes in n trials

For example, suppose we flip a coin 3 times. We can use the formula above to determine the probability of obtaining 0 heads during these 3 flips:

P(X=0) 3C0 * .50 * (1-.5)3-0 = 1 * 1 * (.5)3 = 0.125

The Poisson Distribution

The Poisson distribution describes the probability of experiencing k events during a fixed time interval.

If a random variable X follows a Poisson distribution, then the probability that X = k events can be found by the following formula:

P(X=k) = λk * e– λ / k!

where:

  • λ: mean number of successes that occur during a specific interval
  • k: number of successes
  • e: a constant equal to approximately 2.71828

For example, suppose a particular hospital experiences an average of 2 births per hour. We can use the formula above to determine the probability of experiencing 3 births in a given hour:

P(X=3) = 23 * e– 2 / 3! = 0.18045

Similarities & Differences

The Binomial and Poisson distribution share the following similarities:

  • Both distributions can be used to model the number of occurrences of some event.
  • In both distributions, events are assumed to be independent.

The distributions share the following key difference:

  • In a Binomial distribution, there is a fixed number of trials (e.g. flip a coin 3 times)
  • In a Poisson distribution, there could be any number of events that occur during a certain time interval (e.g. how many customers will arrive at a store in a given hour?)

Practice Problems: When to Use Each Distribution

In each of the following practice problems, determine whether the random variable follows a Binomial distribution or Poisson distribution.

Problem 1: Network Failures

A tech company wants to model the probability that a certain number of network failures occur in a given week. Suppose it’s known that an average of 4 network failures occur each week. Let X be the number of network failures in a given week. What type of distribution does the random variable X follow?

Answer: X follows a Poisson distribution because we’re interested in modeling the number of network failures in a given week and there is no upper limit on the number of failures that could occur. This is not a Binomial distribution because there is not a fixed number of trials.

Problem 2: Shooting Free-Throws

Tyler makes 70% of all free-throws he attempts. Suppose he shoots 10 free-throws. Let X be the number of times Tyler makes a basket during the 10 attempts. What type of distribution does the random variable X follow?

Answer: X follows a Binomial distribution because there is a fixed number of trials (10 attempts), the probability of “success” on each trial is the same, and each trial is independent.

Additional Resources

Binomial Distribution Calculator
Poisson Distribution Calculator

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