Poisson Distribution Calculator

A Poisson experiment examines the number of times an event occurs during a specified interval. The interval could be anything - a unit of time, length, volume, etc. We might, for example, ask how many customers visit a store each day, or how many home runs are hit in a season of baseball.

A Poisson experiment has the following characteristics:

• The average rate of success is known.
• The probability that a single success will occur during a short interval is proportional to the size of the interval.
• The probability that a success will occur within a short interval is independent of successes that occur outside the interval.
• The probability of more than one success occurring within a very short interval is small.

The number of successes in a Poisson experiment is referred to as a Poisson random variable. A Poisson distribution is a probability distribution of a Poisson random variable.

For example, suppose we know that a receptionist receives an average of 1 phone call per hour. We might ask: What is the likelihood that she will get 0, 1, 2, 3, or 4 calls next hour. If we treat the number of phone calls as a Poisson random variable, the various probabilities can be calculated, as shown in the table below.

Taken together, the values for the Poisson random variable and their associated probabilities represent a Poisson distribution.

A Poisson random variable refers to the number of successes in a Poisson experiment.

We might be interested in the number of phone calls received in an hour by a receptionist. Suppose we knew that she received 1 phone call per hour on average. We might ask: What is the likelihood next hour that she will receive 4 phone calls next hour.

If we treated this as a Poisson experiment, then the value of the Poisson random variable would be 4. (And the average rate of success would be 1.)

The average rate of success refers to the average number of successes that occur over a particular interval in a Poisson experiment.

We might be interested in the number of phone calls received in an hour by a receptionist. Suppose she received 1 phone call per hour on average.

If we treated this as a Poisson experiment, then the average rate of success over a 1-hour period would be 1 phone call. Note, however, that our experiment might involve a different unit of time. Suppose we focused on the number of calls during a 30-minute time period. Then, the average rate of success would be 0.5 calls per half hour. Similarly, if we focused on a 2-hour time period, the average rate of success would be 2 calls per 2 hours.

A Poisson probability refers to the probability of getting EXACTLY n successes in a Poisson experiment. Here, n would be a Poisson random variable.

For instance, we might be interested in the number of phone calls received in an hour by a receptionist. Suppose we knew that she received 1 phone call per hour on average. We might ask: What is the likelihood next hour that she will receive EXACTLY 3 phone calls? The probability of getting EXACTLY 3 phone calls in the next hour would be an example of a Poisson probability, denoted by P(X=3). (Note: In this example, the Poisson probability is equal to 0.061.)

A cumulative Poisson probability refers to the probability that the Poisson random variable (X) falls within a certain range. For example, consider the probability of getting AT MOST n successes in a Poisson experiment. Here, n would be a Poisson random variable. And the cumulative Poisson probability would be the probability that n falls within the range of 0 and n.

For instance, we might be interested in the number of phone calls received in an hour by a receptionist. Suppose we knew that she received 1 phone call per hour on average. We might ask: What is the likelihood that she will receive AT MOST 1 phone call next hour? The probability of getting AT MOST 1 phone call in the next hour would be an example of a cumulative Poisson probability.

Note: The cumulative Poisson probability in this example is equal to the probability of getting zero phone calls PLUS the probability of getting one phone call. Thus, the cumulative Poisson probability would equal 0.368 + 0.368 or 0.736. (For details, see the question above: What is a Poisson distribution.)

Notation associated with cumulative Poisson probability is best explained through illustration. The probability of getting LESS THAN 1 phone call is indicated by P(X < 1); the probability of getting AT MOST 1 phone call is indicated by P(X < 1); the probability of getting AT LEAST 1 phone call is indicated by P(X > 1); the probability of getting MORE THAN 1 phone call is indicated by P(X > 1).