So imagine the number of people that show up at a bus stop is Poisson

with a mean of 2.5 per hour.

And you are watching the bus stop for.

Oh, I'm sorry, this is not an example of a Poisson approximation of a binomial.

This is just a regular Poisson.

So we want the number of people that show up

at a bus stop is Poisson with a mean of 2.5 people per hour.

We watch the bus stop for four hours, what's the probability that three or

fewer people show up the whole time?

That is exactly just a Poisson probability.

3, remembering that when we do p for probability distribution,

it does three or less, which is what we want, so we put three in there.

Lambda is 2.5 times the number of hours that we monitored,

4, and that works out to be 1%.

Okay, so see how we used it?

Lambda was the event per unit time, and

t was the number of units of time that we measured.

Okay, so there's your Poisson example.

Let's go through an example of the Poisson approximating the binomial.

We flip a coin with success probability 0.01, so p is small.

We flip it 500 times.

What's the probability of 2 or fewer successes?

So again, pbinom(2, size equals 500, prob equals 0.1.

Here's the exact calculation.

It works out to be 0.12.

Here's the Poisson approximation, ppois(2 lambda = 500 times 0.01.

n times p, and that works out to be 0.1247.

So pretty close.

So that's the Poisson approximation of the binomial.

So here we just showed you exactly how accurate they were in this specific case.

And then, in your regression class, you will actually cover modeling

counts using kind of a Poisson version of regression.

So it's a very convenient model, and it's great for modeling things like rates.

And I would also mention this, Poisson approximation to the binomial is so

common that people don't even acknowledge that they're doing it.

It's just sort of done immediately in a lot of applications,

particularly certain epidemiological applications.

If you're studying something like an infection or something like that that's

rare relative to the size of the population you're studying,

people just automatically use Poisson approximations.