Binomial or Poisson?

The tell is in the wording

Two questions from the same collections desk:

• "Of 200 invoices issued this month, 2 % are typically disputed. How many of the 200 will be disputed?" — a fixed number of trials, each disputed or not, with a constant rate. $X\sim \mathrm{Bin}(200,0.02)$.
• "The desk receives an average of 6 disputed invoices per week. How many will arrive next week?" — a count per interval, with no ceiling and no list of trials. $X\sim \mathrm{Poisson}(6)$.

Binomial when there is an $n$ — a number out of which successes are counted. Poisson when there is a rate per interval of time, length or area, and the count could in principle be anything. Look for "out of" versus "per". A question that gives both an $n$ and a $p$ is binomial; one that gives a rate and an interval is Poisson.

When the binomial is nearly a Poisson

Compute the binomial's chance of exactly three disputes: $P(X=3)=\left(\genfrac{}{}{0ex}{}{200}{3}\right)(0.02{)}^3(0.98{)}^{197}$. The combination is 1,313,400 and $0.98^{197}$ is a long calculator entry; the answer is 0.1963. Now set $\lambda =np=200\times 0.02=4$ and use the Poisson pmf:

$$P(X=3)\approx \frac{e^{-4}{4}^3}{3!}=\frac{0.01832\times 64}{6}=0.1954$$

Off by 0.0009 — one part in two hundred. When $n$ is large and $p$ small, a binomial count behaves like a Poisson count with rate $np$, because a rare event over many trials is what a Poisson process is. The approximation was invented for hand computation and the paper still asks for it.

The rule the course uses

IBS1 states three different thresholds for when the approximation may be used. The course adopts one, as a convention:

> Use the Poisson approximation to $\mathrm{Bin}(n,p)$, with $\lambda =np$, when $n\ge 20$ and $p\le 0.05$.

Say it is a convention if the paper asks; other texts use $n\ge 100$ and $np\le 10$, and they are not wrong. The mean is exactly right either way ($np=\lambda$); the variance is slightly overstated ($\lambda$ versus $np(1-p)=3.92$), which is why the approximation wants $p$ small.

Six scenarios to classify

1. Errors among 40 audited claims — binomial, $n=40$.
2. Calls to the help desk per hour — Poisson, rate per hour.
3. Defective units in a shipment of 500 at 1 % — binomial, and a candidate for the approximation ($\lambda =5$).
4. Cheques bouncing among the 8 cheques deposited today — binomial, $n=8$.
5. Flaws per square metre of fabric — Poisson, rate per area.
6. Customers until the first complaint — neither: no fixed $n$, no interval.

Setting up, not solving

A three-mark exam question often stops at the setup: name the distribution and why, rescale the rate, write $P(\cdot )$ as an expression. For "more than 2 disputed invoices on a given day", with 6 per five-day week: $\lambda =6/5=1.2$ per day, $X\sim \mathrm{Poisson}(1.2)$, $P(X>2)=1-[p(0)+p(1)+p(2)]=1-e^{-1.2}(1+1.2+0.72)$. Evaluated, 0.1205 — but the marks are in the three lines before the number.