Poisson Distribution Calculator

How to Use the Poisson Distribution Calculator

This calculator allows you to calculate exact and cumulative Poisson probabilities when you know the average number of events expected in a given interval.

To use the calculator:

1. Enter the average rate, λ.
2. Enter the number of occurrences, x. For a probability between two values, enter the lower and upper number of occurrences.
3. Select the probability you want to find.
4. Choose the number of decimal places for the final answer.
5. Click Calculate.

The calculator returns the correct probability you want to find and shows you exactly how the probability was computed using your data values. It will also provide you with the mean and standard deviation of the Poisson distribution you’re working on.

Therefore, with the calculator, you can compute these Poisson probabilities:

• Probability that exactly x events occur, P(X = x)
• Probability that at most x events occur, P(X ≤ x)
• Probability that fewer than x events occur, P(X < x)
• Probability that at least x events occur, P(X ≥ x)
• Probability that more than x events occur, P(X > x)
• Probability that between a and b events occur, P(a ≤ X ≤ b)

What Is a Poisson Distribution?

The Poisson distribution is a discrete probability distribution used to model the number of times an event occurs within a fixed interval when the events occur independently, and the average event rate remains reasonably constant.

The interval does not necessarily have to be time. Depending on the problem, it could represent time, distance, area, volume, or another fixed amount of exposure.

For example, a Poisson distribution may be used to model the number of:

• customers arriving at a store per hour;
• calls received by a help desk per minute;
• defects found on a sheet of material;
• accidents occurring at an intersection per month;
• website errors occurring during a fixed time period.

The random variable X represents the number of occurrences. Thus, its possible values are 0, 1, 2, 3, and so on. The Poisson probability formula and cumulative distribution are standard ways of calculating these probabilities.

When Should You Use the Poisson Distribution?

A Poisson distribution is appropriate when you are interested in how many times an event occurs within a fixed interval rather than the number of successes from a fixed number of trials.

In general, the Poisson model assumes that:

• X counts the number of events in a fixed interval;
• events occur independently;
• the average rate of occurrence remains approximately constant over the interval;
• X takes whole-number values beginning at 0.

For example, if a hospital receives an average of 4 emergency calls per hour and you want the probability of receiving exactly 6 calls in the next hour, a Poisson model may be appropriate.

However, if a problem instead gives a fixed number of trials and a probability of success for each trial, you will usually need the Binomial Distribution Calculator rather than a Poisson distribution.

Poisson Distribution Formula

The Poisson probability formula is:

Where:

• P(X = x) is the probability of exactly x occurrences;
• λ is the average number of occurrences in the interval;
• x is the number of occurrences you want to find;
• x! is the factorial of x.

The formula allows you to find the exact probability of x occurrences. Therefore, to find cumulative probabilities, you’ll need to compute several exact Poisson probabilities and add them together.

How to Calculate Poisson Probability

To calculate Poisson probability manually, follow these steps:

• Identify the average rate, λ, and the number of occurrences involved in the question.
• For exact probability, substitute λ and x directly into the Poisson probability formula. However, for a cumulative probability, calculate the required exact probabilities and add them.

Tip. When working with cumulative Poisson probabilities such as at least or more than, you can often simplify the calculation using the complement rule. For instance, P(X>5) = 1- P(X ≤ 4).

Example 1: Exact Poisson Probability

A university IT help desk in the United States receives an average of 3 urgent support requests per hour. Assume the number of requests follows a Poisson distribution. What is the probability that the help desk receives exactly 5 urgent requests in the next hour?

Solution

From the question, we know that:

• Average rate, λ = 3
• Number of occurrences, x = 5

Since we need the probability of exactly 5 requests, we need to find P(X = 5)

Recall. The Poisson probability formula is: P(X = x) = (e^−λ × λ^x) / x!

Substituting λ = 3 and x = 5 gives:

P(X = 5) = (e^-3 × 3⁵) / 5!

= 12.0983/120

= 0.1008

This implies that the probability that the help desk receives exactly 5 urgent support requests in the next hour is approximately 0.1008.

Example 2: Cumulative Poisson Probability

A US urgent care clinic receives an average of 2.5 walk-in patients every 30 minutes. Assume the number of arrivals follows a Poisson distribution. What is the probability that at least 4 patients arrive during the next 30 minutes?

Solution

From the question, we know that:

• Average rate, λ = 2.5
• Number of occurrences, x = 4

“At least 4” means 4, 5, 6, and all higher possible numbers of arrivals. Thus, we need to find P(X ≥ 4)

Therefore, instead of adding an infinite number of probabilities, we can use the complement rule.

Using the complement rule, P(X ≥ 4) = 1 − P(X ≤ 3)

Therefore, we first need to find:

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Using the Poisson probability formula:

P(X = 0) = (e^-2.5 × 2.5⁰) / 0!

= 0.08208

P(X = 1) = (e^-2.5 × 2.5¹) / 1!

= 0.20521

P(X = 2) = (e^-2.5 × 2.5²) / 2!

=0.513031/2

= 0.25652

P(X = 3) = (e^-2.5 × 2.5³) / 3!

= 1.282578/6

= 0.21376

Since we have the exact probabilities, P(X ≤ 3) = 0.08208 + 0.20521 + 0.25652 + 0.21376

= 0.75757

Recall. P(X ≥ 4) = 1 − P(X ≤ 3) using the complement rule.

Therefore, P(X ≥ 4) = 1 − 0.75757

= 0.24243

This implies that the probability that at least 4 walk-in patients arrive during the next 30 minutes is approximately 0.2424.

Mean and Standard Deviation of a Poisson Distribution

Besides the requested probability, the calculator also reports the mean and standard deviation of the Poisson distribution.

For a Poisson distribution, the mean and variance are equal to λ. Therefore, the standard deviation of a Poisson distribution is √λ. In other words:

• Mean, μ = λ
• Variance, σ² = λ
• Standard deviation, σ = √λ.

Common Mistakes When Calculating Poisson Probability

Avoid these common errors:

• Using the wrong λ. Make sure the average rate matches the interval in the question.
• Confusing exactly with at most. P(X = 3) includes only 3, while P(X ≤ 3) includes 0, 1, 2, and 3.
• Confusing at least with more than. P(X ≥ 3) includes 3, but P(X > 3) does not.
• Entering a decimal for x. The number of occurrences must be a whole number such as 0, 1, 2, or 3.
• Using the Poisson distribution for a fixed number of trials. If n and p are given, a binomial distribution may be more appropriate.
• Forgetting the complement rule. Upper-tail probabilities are often easier to calculate as 1 minus a cumulative lower-tail probability.

Frequently Asked Questions