Difficulty: Intermediate | Prerequisites: random variables, PMF, expected value, variance, and some familiarity with the binomial distribution (earlier in Ch. 5).
The Poisson distribution handles a different kind of counting problem from the binomial. Instead of asking "how many successes in n trials," it asks "how many events occur in a fixed interval of time, space, or volume?" It appears whenever events happen independently, at a known average rate, and are rare enough that two occurrences in the same tiny instant are negligible. It is widely used in fields from telecommunications (calls per hour) to biology (mutations per gene) to insurance (claims per month).
The Poisson distribution counts the number of events in a fixed interval when those events happen independently at a constant average rate λ. Its PMF, mean, and variance all revolve around λ: the mean equals λ, and so does the variance.
Poisson distribution
A discrete probability distribution that gives the probability of a certain number of events occurring in a fixed interval (of time, distance, area, or volume) when the events happen independently at a constant average rate. Think of it as the go-to model for "how many times does something happen in this window?"
λ (lambda)
The average number of events per interval. It is the single parameter of the Poisson distribution and determines both the mean and the variance. In simple terms, λ is the rate: if a call centre receives an average of 4 calls per minute, λ = 4.
Fixed interval
The predetermined window over which you are counting events. It could be one hour, one square metre, one page of text, or any other defined span. The interval must be set before you observe.
Independence (Poisson context)
One event occurring does not change the probability of another event occurring in the same interval. This is what separates Poisson-eligible events from clustered or dependent ones.
Rare-event condition
The probability of more than one event occurring in a very small sub-interval is negligible. This ensures events do not bunch up at exactly the same instant.
Use it when you are counting events in a fixed interval and all of the following hold:
The average rate λ is known (or can be estimated).
Events are independent of one another.
The rate is proportional to the size of the interval (double the interval, double the expected count).
The probability of two or more events occurring at exactly the same instant is essentially zero.
P(X = x) = (e⁻λ × λˣ) / x!, where x = 0, 1, 2, …
Unlike the binomial, there is no upper limit on x in principle (though very large values have vanishingly small probabilities).
µ = λ. The mean number of events per interval is simply the rate parameter.
σ² = λ. A distinctive feature of the Poisson: the variance equals the mean.
σ = √λ.
This equality of mean and variance is a useful diagnostic. If you have count data and the sample mean and sample variance are roughly equal, that is a hint the Poisson model may fit.
When n is large and p is very small (so that np = λ stays moderate), the binomial distribution is well approximated by the Poisson with λ = np. This is sometimes called the "law of rare events." In practice, many textbooks suggest the approximation works well when n ≥ 20 and p ≤ 0.05.
P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad x = 0, 1, 2, \ldots\mu = \lambda\sigma^2 = \lambda\sigma = \sqrt{\lambda}The Poisson distribution models the number of emails arriving per hour, the number of typos per page, the number of car accidents at an intersection per month, and the number of radioactive decay events detected per second. Anywhere you count independent events over a fixed span, the Poisson is likely the right starting model.
Students sometimes confuse the Poisson with the binomial. The binomial counts successes in a fixed number of trials; the Poisson counts events in a fixed interval with no upper limit on the count.
Assuming the Poisson applies when events are not independent. If one event triggers another (e.g., aftershocks following an earthquake), the Poisson does not fit.
Forgetting that for the Poisson, the mean and variance are both equal to λ. If a problem gives you separate values for the mean and variance that differ substantially, the Poisson is not the right model.
⚠️ Be ready to identify whether a scenario calls for a binomial or a Poisson. The key question: are you counting successes in a fixed number of trials (binomial) or events in a fixed interval (Poisson)?
⚠️ Know the PMF formula and be comfortable plugging in λ and x. Exam questions often ask for P(X = 0), P(X = 1), or P(X ≤ 2).
⚠️ The fact that µ = σ² = λ is a frequently tested property.
True or false: the Poisson distribution has an upper limit on the number of events.
False. In theory x can be any non-negative integer.
Fill in the blank: for a Poisson random variable, the variance equals ___.
λ (the same as the mean).
True or false: the Poisson is a good model when events tend to cluster together.
False. Events must be independent.
Fill in the blank: the Poisson approximation to the binomial works when n is ___ and p is ___.
n is large; p is very small.
True or false: P(X = 0) for a Poisson distribution equals e⁻λ.
True. Plugging x = 0 into the formula gives e⁻λ × λ⁰ / 0! = e⁻λ.
Q: A bookshop receives an average of 3 customer complaints per week. What is the probability of receiving exactly 5 complaints in a given week?
A: λ = 3. P(X = 5) = e⁻³ × 3⁵ / 5! = e⁻³ × 243 / 120 ≈ 0.1008.
Q: Using the same scenario, what is P(X = 0)?
A: P(X = 0) = e⁻³ × 3⁰ / 0! = e⁻³ ≈ 0.0498.
Q: A radioactive source emits particles at a rate of 2 per second. What are the mean and standard deviation of the number of particles emitted in one second?
A: µ = λ = 2. σ = √2 ≈ 1.414.
Q: A hospital emergency department sees an average of 10 patients per hour. If you double the observation window to two hours, what is λ for the two-hour interval?
A: λ = 10 × 2 = 20. The rate scales proportionally with the interval.
The Poisson distribution bridges the discrete distributions in this chapter and the continuous distributions ahead. In particular, the time between Poisson events follows an exponential distribution (a continuous model covered later). The Poisson also serves as an approximation to the binomial under rare-event conditions, which ties these two chapter topics together. Understanding λ as both the mean and variance prepares you for hypothesis tests and goodness-of-fit tests in later chapters.
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