Poisson Distribution: When Events Are Random

The Poisson distribution is one of the most useful probability distributions in statistics. It models the number of times an event occurs in a fixed interval of time or space, given that the events happen independently at a known average rate. From the number of calls arriving at a call centre per hour to the number of typos on a printed page, the Poisson distribution appears everywhere in real-world data analysis.

This guide explains what the Poisson distribution is, when to use it, how to calculate probabilities with its formula, and works through several practical examples. We also compare it with the binomial distribution and cover cumulative probabilities.

What Is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that gives the probability of a certain number of events occurring in a fixed interval, provided:

• Events occur independently of one another.
• The average rate (denoted ) is constant.
• Two events cannot occur at exactly the same instant.
• The probability of an event in a very small interval is proportional to the length of that interval.

The distribution is named after the French mathematician Simeon Denis Poisson, who published it in 1837.

The Probability Mass Function (PMF)

The probability that exactly events occur is:

where:

• is the average number of events per interval (also called the rate parameter)
• is the number of occurrences (0, 1, 2, 3, ...)
• is Euler's number
• is the factorial of

Mean and Variance

One of the distinctive features of the Poisson distribution is that its mean and variance are both equal to :

This means the standard deviation is . If you observe data where the sample mean and sample variance are approximately equal, that is a strong hint that the Poisson model may be appropriate.

Worked Example 1: Call Centre Arrivals

A call centre receives an average of 8 calls per hour. What is the probability of receiving exactly 5 calls in a given hour?

Here and . Applying the PMF:

Using :

There is approximately a 9.16% chance of receiving exactly 5 calls in that hour. This makes intuitive sense: 5 calls is somewhat below the average of 8, so the probability is moderate but not negligible.

Worked Example 2: Website Server Crashes

A web server experiences an average of 2 crashes per month. What is the probability of having zero crashes in a given month?

Here and :

There is roughly a 13.53% chance of a crash-free month. What about 3 or more crashes?

Computing each term:

There is about a 32.33% chance of 3 or more crashes per month. A systems administrator would likely want to reduce before this risk level is acceptable.

Worked Example 3: Typos Per Page

A publisher finds that a particular author averages 1.5 typos per page. What is the probability that a randomly selected page has exactly 2 typos?

With and :

About 25.1% of pages will contain exactly 2 typos. The probability of a clean page (zero typos) is:

So roughly 22.3% of pages are error-free. Together, these calculations help the publisher plan how much proofreading resource to allocate.

Cumulative Probabilities

Often you need the probability of or fewer events, known as the cumulative distribution function (CDF):

For “at least” probabilities, use the complement:

We used this approach in Example 2 when calculating . For large values of , summing many terms by hand is tedious, which is where a calculator becomes invaluable.

Poisson vs Binomial: When to Use Which

The binomial and Poisson distributions are closely related. The binomial models the number of successes in a fixed number of independent trials, each with the same probability of success. The Poisson models the number of events in a continuous interval.

Use the binomial when: you know the number of trials and the probability of success on each trial.

Use the Poisson when: you know the average rate but the number of “trials” is not well defined (e.g., calls per hour, defects per metre).

There is a famous approximation: when is large and is small, the binomial distribution is well approximated by the Poisson with . A common rule of thumb is that the approximation works well when and .

Worked Example 4: Poisson Approximation to the Binomial

A factory produces 1,000 items per day. Each item has a 0.002 probability of being defective. What is the probability of finding exactly 3 defective items in a day?

The exact binomial calculation uses and , which involves enormous factorials. Instead, use the Poisson approximation with :

There is about an 18% chance of exactly 3 defective items. The Poisson approximation here is extremely close to the exact binomial value (which is approximately 0.1806), saving significant computational effort.

Properties and Special Cases

Sum of Independent Poisson Variables

If and are independent, then:

This is useful when combining counts from multiple independent sources. For instance, if two departments receive calls at rates 5 and 3 per hour respectively, the total call rate is Poisson with .

Scaling the Interval

If events occur at rate per unit time, then in an interval of length the number of events follows . For instance, if a shop averages 12 customers per hour (), the number of customers in a 15-minute window follows .

Connection to the Exponential Distribution

If the number of events follows a Poisson distribution, then the time between consecutive events follows an exponential distribution with rate . This link is fundamental in queueing theory and reliability engineering.

Common Mistakes to Avoid

• Mismatching the interval. If is given per hour but you need the probability over 30 minutes, use , not .
• Using the Poisson when events are not independent. If one event increases or decreases the chance of the next, the Poisson model does not apply.
• Confusing PMF and CDF. The formula gives the probability of exactly events. For “at most” or “at least” questions, you need to sum multiple terms or use the complement.
• Forgetting that . The case simplifies to , which students sometimes miss.

Frequently Asked Questions

What is the Poisson distribution in simple terms?

The Poisson distribution tells you the probability of a given number of events happening in a fixed period of time (or fixed region of space), when those events occur randomly and independently at a known average rate. It answers questions like “what is the chance of getting exactly 4 emails in the next hour if I usually receive 6 per hour?”

When should I use the Poisson distribution?

Use it when you are counting the number of occurrences of an event over a fixed interval, the events are independent, the average rate is known, and two events cannot happen at the same instant. Classic examples include arrivals at a service point, radioactive decay events, accidents at a junction, and defects in manufactured goods.

Why are the mean and variance equal in a Poisson distribution?

This is a mathematical consequence of the PMF formula. It can be proven by computing and directly from the definition. In practical terms, it means that as the average rate increases, the spread of the distribution increases at the same pace. If your data shows the variance significantly exceeding the mean (overdispersion), a negative binomial model may be more appropriate.

How do I calculate Poisson probabilities for large values of lambda?

For large (say above 20), computing and directly can cause overflow. Use logarithms: , then exponentiate. Alternatively, for very large , the Poisson distribution is well approximated by a normal distribution with mean and variance .

Can the Poisson distribution be used for non-time intervals?

Absolutely. The “interval” can be any fixed region: a length of wire (defects per metre), an area of fabric (flaws per square metre), a volume of liquid (bacteria per litre), or even abstract quantities like mutations per gene. The key requirement is a constant average rate over the region, with independent occurrences.