The Poisson DistributionThe Poisson distribution models the number of events that occur in a fixed interval of time or space when those events happen at a known average rate, independently of one another and one at a time. It is the typical distribution of counts: calls per hour, accidents per day, defects per metre of fabric, typos per page.The Poisson Distribution FormulaThe probability of exactly \( k \) events occurring in the interval is:\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}, \quad k = 0, 1, 2, \dots \]Where:\( \lambda \): average number of occurrences in the interval considered;\( k \): number of events whose probability we want;\( e \): Euler's number (approximately 2.71828);\( k! \): factorial of \( k \).Two properties are very useful in practice:The mean and the variance are both equal to \( \lambda \). If your count data have a variance much larger than the mean, the Poisson is probably not the right model.The rate scales with the length of the interval. If there are, on average, 5 calls per hour, the mean is 10 in 2 hours and 2.5 in 30 minutes. Before applying the formula, adjust \( \lambda \) to the interval in the question.A website receives, on average, 2 orders per minute, and orders follow a Poisson distribution. What is the probability that no order arrives in a given minute?Answer: 13.53% — with k = 0 the formula becomes e to the minus 2, times 2 to the power 0, divided by 0!, that is, just e to the minus 2, which is about 0.1353. The 27.07% option is the probability of exactly 2 orders, not of none.Real-Life ExamplesExample 1: Phone callsSituation: a call center receives, on average, 5 calls per hour.Question: what is the probability of receiving exactly 3 calls in one hour?Solution:\[ \lambda = 5, \quad k = 3 \]\[ P(X = 3) = \frac{e^{-5} \cdot 5^3}{3!} = \frac{0.006738 \cdot 125}{6} \approx 0.1404 \quad (14.04\%) \]Example 2: Accidents on a highwaySituation: on a highway there are, on average, 2 accidents per day.Question: what is the probability of exactly 4 accidents in one day?Solution:\[ \lambda = 2, \quad k = 4 \]\[ P(X = 4) = \frac{e^{-2} \cdot 2^4}{4!} = \frac{0.1353 \cdot 16}{24} \approx 0.0902 \quad (9.02\%) \]Example 3: Defects on a production lineSituation: a factory produces, on average, 1 defective part every 10 hours.Question: what is the probability of finding exactly 2 defective parts in 20 hours?Solution: first we adjust the rate to the interval in the question. If there is 1 defect every 10 hours, in 20 hours we expect 2 defects:\[ \lambda = 1 \cdot \frac{20 \text{ hours}}{10 \text{ hours}} = 2 \]\[ P(X = 2) = \frac{e^{-2} \cdot 2^2}{2!} = \frac{0.1353 \cdot 4}{2} \approx 0.2707 \quad (27.07\%) \]The chart below shows how the shape of the distribution changes as \( \lambda \) grows: with \( \lambda = 1 \) the probabilities are concentrated at 0 and 1; with \( \lambda = 10 \) the curve becomes wider and almost symmetric around 10.Poisson distribution for λ = 1, λ = 4 and λ = 10On a highway there are, on average, 2 accidents per day. What is the probability of at least 1 accident in a day?Answer: 86.47% — at least 1 is the complement of none: 1 minus P(X = 0) = 1 minus e to the minus 2 = 1 minus 0.1353 = 0.8647. The 72.93% option comes from also subtracting P(X = 1), which would give at least 2 accidents.Historical Background of the Poisson DistributionThe Poisson distribution is named after the French mathematician Siméon Denis Poisson (1781–1840), who published it in 1837 in his book Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Poisson was studying the probability of verdicts in trials, that is, relatively rare events within large populations. An equivalent form of the formula had already appeared in a 1711 work by Abraham de Moivre, but the name that stuck was Poisson's.The distribution arose as an approximation to the binomial distribution when the number of trials \( n \) is large and the probability of success \( p \) is small, keeping the product \( \lambda = n \cdot p \) constant. For example, with \( n = 1000 \) and \( p = 0.002 \) we have \( \lambda = 2 \): the exact binomial gives \( P(X = 3) \approx 18.06\% \) and the Poisson gives \( 18.04\% \), with a much simpler calculation.A print shop prints 1,000 flyers and each one has a probability of 0.002 of being defective, independently of the others. Using the Poisson approximation, what are λ and the probability of exactly 3 defective flyers?Answer: λ = 2 and P ≈ 18.04% — λ = n times p = 1000 times 0.002 = 2, and P(X = 3) = e to the minus 2, times 8, divided by 6, which is about 0.1804. The option with λ = 3 confuses the desired number of events (k) with the average rate.When to Use Poisson and When to Use Binomial?The choice between the binomial and the Poisson distribution depends on the characteristics of the problem. Both count events, but in different contexts.When to Use the Binomial DistributionThe binomial distribution is used when:There is a fixed number of trials (\( n \)): the experiment has a known number of repetitions.There are two possible outcomes: each trial results in success or failure.The probability is constant (\( p \)): the chance of success is the same in every trial.Trials are independent: the result of one does not affect the others.Examples of application:Tossing a coin \( n \) times and counting the number of heads.Testing \( n \) products and counting how many are defective.Taking \( n \) penalty kicks and counting how many goals are scored.When to Use the Poisson DistributionThe Poisson distribution is used when:There is no fixed number of trials: we count occurrences in a continuous interval (time, area, length, volume), not successes in \( n \) trials.The average rate (\( \lambda \)) is known and constant throughout the interval.Events are independent: one occurrence does not change the chance of the others.Events happen one at a time: the chance of two events at exactly the same instant is negligible.You will often read that the Poisson is for rare events. That is true in the sense that each small piece of the interval has a small chance of containing an event; the total in the interval can be large, like the 20 cars per minute in exercise 8.Examples of application:Number of calls received by a call center per hour.Number of accidents on a highway per day.Number of typos on a page of text.Number of customers arriving at a restaurant per minute.Which of these situations is modelled by the binomial distribution rather than the Poisson?Answer: Number of heads in 20 tosses of a coin — there is a fixed number of trials (20), two possible outcomes and a constant probability. In the other three there is no fixed number of trials: we count occurrences over an interval of time or length, which is Poisson territory.A store counted customers per hour over several days and obtained a mean of 4 and a variance of 12. Is the Poisson distribution a good model for this count?Answer: No, because in the Poisson the variance equals the mean, and here it is three times larger — this overdispersion indicates that customers do not arrive independently (they come in groups or there are peak hours). The Poisson does not require a mean below 1; the third option is exactly the mistake of applying the formula without checking the data.ExercisesPhone calls: a call center receives, on average, 4 calls per hour. What is the probability of receiving exactly 6 calls in one hour?Highway accidents: on a highway there are, on average, 3 accidents per day. What is the probability of exactly 5 accidents in one day?Defects on a production line: a factory produces, on average, 2 defective parts every 8 hours. What is the probability of finding exactly 3 defective parts in 24 hours?Customer arrivals: a restaurant receives, on average, 10 customers per hour. What is the probability of receiving exactly 12 customers in one hour?Emails received: an employee receives, on average, 8 emails per hour. What is the probability of receiving exactly 10 emails in one hour?System failures: a system fails, on average, once a day. What is the probability that it fails exactly 2 times in one day?Births in a hospital: a hospital records, on average, 5 births per day. What is the probability of exactly 7 births in one day?Cars passing a toll booth: a toll booth records, on average, 20 cars per minute. What is the probability that exactly 25 cars pass in one minute?Typing errors: a typist makes, on average, 3 errors per page. What is the probability of making exactly 5 errors on one page?Rain in a city: in a city it rains, on average, 4 days a month. Using the Poisson, what is the probability that it rains on exactly 6 days in a month?Answer Key10.42% (\( \lambda = 4,\ k = 6 \))10.08% (\( \lambda = 3,\ k = 5 \))8.92% (in 24 hours, \( \lambda = 2 \cdot 24/8 = 6 \); \( k = 3 \))9.48% (\( \lambda = 10,\ k = 12 \))9.93% (\( \lambda = 8,\ k = 10 \))18.39% (\( \lambda = 1,\ k = 2 \))10.44% (\( \lambda = 5,\ k = 7 \))4.46% (\( \lambda = 20,\ k = 25 \))10.08% (\( \lambda = 3,\ k = 5 \))10.42% (\( \lambda = 4,\ k = 6 \)). Note: since a month has at most 30 or 31 days and each day either has rain or not, this case is strictly binomial (\( n = 30,\ p = 4/30 \)), which gives about 10.76%. The Poisson is a reasonable approximation here, but not an exact one.
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