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.
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.