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
- n: number of trials
- k: number of successes
- p: probability of success on a given trial
- nCk: the 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!
- 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.