The Poisson distribution is a probability model for counting how many times an event happens in a fixed interval of time, space, area, or exposure. It is useful when events are relatively rare, occur independently, and have a steady average rate. Common examples include phone calls per hour, printing errors per page, radioactive decays per second, or accidents at an intersection per month.
It matters because it turns an average rate into a full set of probabilities for 0, 1, 2, 3, and more events.
The key parameter is lambda, written λ, which represents the expected number of events in the interval. Once λ is known, the probability of observing exactly k events is P(X = k) = e^-λ λ^k / k!. The distribution is often right skewed when λ is small, but it becomes more symmetric as λ gets larger.
In applications, the Poisson model helps estimate risk, plan staffing, detect unusual event counts, and compare observed counts with what would be expected by chance.
Understanding Statistics: The Poisson Distribution
A useful way to picture this model is to divide an interval into many tiny equal pieces. In each tiny piece, an event has only a small chance of occurring. Most pieces contain nothing.
A few contain one event. The chance of two events in the same tiny piece is so small that it can be ignored. When these small chances stay similar from piece to piece, the total count has a Poisson pattern.
This picture explains why the model is built for discrete counts. A result such as two point four calls has no meaning, but an average of two point four calls over many hours does have meaning.
The average rate must match the interval being studied. Suppose a radiation detector records an average of two particle decays each minute. For a one minute observation, the expected count is two.
For five minutes, it is ten, provided the detector and source behave consistently. A longer interval usually has a larger expected count and a wider spread of possible results. Counts still vary naturally from one interval to the next.
With an expected count of two, no decays in one minute is not surprising. It occurs about once in every seven minutes on average. Exactly two decays is the single most likely count, but three or more decays together are possible often enough that they should not be treated as an error.
The assumptions matter more than memorising a calculation rule. Independence means one event does not make another event more or less likely in the next moment. This can fail when events arrive in bursts.
Messages sent during a school emergency may cluster because one message causes replies. A steady rate can fail across a whole day. Customer arrivals may be slow in the morning and busy after school.
In these cases, separate time periods may need separate rates. Another warning sign comes from data.
If repeated counts vary much more than their average suggests, clustering or changing conditions may be present. If they vary much less, there may be a limit on the number of possible events, such as a fixed number of seats or machines.
When solving problems, first state exactly what counts as one event. Then identify the interval and make the units agree before finding the expected count. A rate per hour must be converted carefully for a ten minute interval.
Read probability wording closely. Exactly four means one count only. At most four includes zero through four.
At least four includes four plus every larger count. Large tails are often easiest to find by subtracting smaller probabilities from one. In an investigation, collect many equal intervals, calculate the sample average, then compare the observed counts with the expected pattern.
A few unusual intervals can happen by chance. A repeated pattern of unusually high or low counts is stronger evidence that the Poisson assumptions need checking.
Key Facts
- Poisson probability formula: P(X = k) = e^-λ λ^k / k!, where k = 0, 1, 2, ...
- λ is the mean number of events in the chosen interval, so E(X) = λ.
- For a Poisson random variable, the variance is also λ, so Var(X) = λ and σ = sqrt(λ).
- Use the Poisson distribution for counts in a fixed interval when events are independent and the average rate is constant.
- If the rate is r events per unit and the interval length is t units, then λ = rt.
- The probabilities over all possible counts add to 1: P(X = 0) + P(X = 1) + P(X = 2) + ... = 1.
Vocabulary
- Poisson distribution
- A probability distribution that models the number of independent events occurring in a fixed interval when the average rate is known.
- Lambda
- The parameter λ that represents the average or expected number of events in the interval being studied.
- Random variable
- A variable whose value depends on the outcome of a random process, such as the number of calls received in one hour.
- Independence
- A condition where one event occurring does not change the probability of another event occurring.
- Factorial
- The product of all positive integers up to a given number, such as 4! = 4 × 3 × 2 × 1.
Common Mistakes to Avoid
- Using the Poisson distribution for measurements instead of counts, which is wrong because Poisson models whole-number event counts like 0, 1, 2, not continuous values like height or time.
- Forgetting to adjust λ to match the interval, which is wrong because a rate of 6 calls per hour gives λ = 3 for a half hour, not λ = 6.
- Using the formula with a negative or decimal value of k, which is wrong because k must be a nonnegative whole number count.
- Assuming Poisson events can influence each other, which is wrong because the model requires independence and a constant average rate across the interval.
Practice Questions
- 1 A help desk receives an average of 4 calls per hour. Assuming calls follow a Poisson distribution, what is the probability of receiving exactly 2 calls in one hour?
- 2 A website averages 10 form submissions per day. What is the probability of receiving exactly 3 submissions in a 6-hour period, assuming the rate is constant throughout the day?
- 3 A student wants to model the number of cars passing a checkpoint in 10 minutes using a Poisson distribution. Explain what conditions should be checked before using the model.