Three per hour is six per two hours. Stretch the window and watch both the mean and the variance follow, because for a Poisson they are the same number.
Motorbike taxis arrive at a BTS station at random, on average 3 an hour. The top strip is real arrivals on a timeline. Stretch the shaded window and watch the distribution of the count inside it.
The rate is fixed at 3 an hour. λ is not the rate: it is the rate times the length of the window.
If X ~ Po(λ) then P(X = x) = e−λ λxx! for x = 0, 1, 2, … with no upper limit, and E(X) = Var(X) = λ.
It is not binomial. A binomial counts successes out of a fixed n, so it stops at n. A Poisson counts events in an interval with no n at all, so there is no largest possible value. If a question gives you “out of 50 people” it is binomial; if it gives you “per hour”, “per page” or “per km” it is Poisson.
Adding independent Poissons adds the λs. If one till serves Po(4) customers an hour and another Po(5), the two together are Po(9). This is why scaling the interval works at all: two hours is just one hour plus one hour.
The rate is not constant. Motorbike taxis peak at 8am. Across a whole day Poisson will understate the quiet hours and the busy ones at once.
Events arrive in clumps. Goals, accidents in one heavy downpour, retweets. Poisson requires independence, and clumping shows up as a variance larger than the mean.
So there is a test you can do. Compute the sample mean and the sample variance. If they are far apart, Poisson is a poor fit, and saying so earns the mark.
1. Potholes occur at 2 per 100 m on a road. Find λ for a 250 m stretch.
2. X ~ Po(9). State the standard deviation.
3. X ~ Po(1). Find P(X = 0), to three significant figures.
4. A sample of counts has mean 4.1 and variance 11.8. What should you conclude?
Scaling λ to the interval the question asks about, before touching the calculator. This single step is where most of the lost marks are.
Reading inequalities correctly. “At least 2” is 1 − P(X ≤ 1), and there is no upper limit to subtract from at the top end.
Justifying the model when asked: constant average rate, independent events, and a mean roughly equal to the variance.
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