The Poisson distribution

Covers AS91586

When to use the Poisson distribution

The Poisson distribution models the number of events in a fixed interval of time or space, when:
- events happen at a constant average rate;
- events occur independently;
- two events cannot occur at exactly the same instant.
Typical examples: calls to a call centre per hour, flaws per metre of cable, cars past a point per minute.

The Poisson formula

With mean rate $\lambda$ (lambda) per interval, the probability of exactly $k$ events is:

$P(X = k) = e^{-\lambda}\,\frac{\lambda^k}{k!}$

For the Poisson distribution, the mean equals the variance, both equal to $\lambda$ .

Matching the interval

$\lambda$ must match the interval in the question. If the rate is 3 per hour but you want a probability for half an hour, use $\lambda = 1.5$ .

Worked ExampleA Poisson probability

A call centre receives calls at an average rate of 4 per hour. Find the probability of exactly 2 calls in a given hour.

Step 1 — Identify $\lambda$ and $k$

The interval is one hour, so $\lambda = 4$ ; we want $k = 2$ .

Step 2 — Substitute into the formula

$P(X = 2) = e^{-4}\,\frac{4^2}{2!} = e^{-4}\,\frac{16}{2} = 8e^{-4}$

Step 3 — Evaluate

$8e^{-4} \approx 8 \times 0.0183 \approx 0.147$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Faults occur at an average of 3 per metre of wire. State the mean and variance of the Poisson distribution for faults per metre.

Merit

Emails arrive at an average of 5 per hour. Find the probability of exactly 3 emails in an hour.

Excellence

Cars pass a point at 6 per minute on average. Find the probability that no cars pass in a 30-second interval.

The Poisson distribution

Covers AS91586

When to use the Poisson distribution

The Poisson distribution models the number of events in a fixed interval of time or space, when:
- events happen at a constant average rate;
- events occur independently;
- two events cannot occur at exactly the same instant.
Typical examples: calls to a call centre per hour, flaws per metre of cable, cars past a point per minute.

The Poisson formula

With mean rate $\lambda$ (lambda) per interval, the probability of exactly $k$ events is:

$P(X = k) = e^{-\lambda}\,\frac{\lambda^k}{k!}$

For the Poisson distribution, the mean equals the variance, both equal to $\lambda$ .

Matching the interval

$\lambda$ must match the interval in the question. If the rate is 3 per hour but you want a probability for half an hour, use $\lambda = 1.5$ .

Worked ExampleA Poisson probability

A call centre receives calls at an average rate of 4 per hour. Find the probability of exactly 2 calls in a given hour.

Step 1 — Identify $\lambda$ and $k$

The interval is one hour, so $\lambda = 4$ ; we want $k = 2$ .

Step 2 — Substitute into the formula

$P(X = 2) = e^{-4}\,\frac{4^2}{2!} = e^{-4}\,\frac{16}{2} = 8e^{-4}$

Step 3 — Evaluate

$8e^{-4} \approx 8 \times 0.0183 \approx 0.147$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Faults occur at an average of 3 per metre of wire. State the mean and variance of the Poisson distribution for faults per metre.

Merit

Emails arrive at an average of 5 per hour. Find the probability of exactly 3 emails in an hour.

Excellence

Cars pass a point at 6 per minute on average. Find the probability that no cars pass in a 30-second interval.

The Poisson distribution

When to use the Poisson distribution

The Poisson formula

Matching the interval

Step 1 — Identify λ\lambdaλ and kkk

Step 2 — Substitute into the formula

Step 3 — Evaluate

Test yourself

Practice by grade

The Poisson distribution

When to use the Poisson distribution

The Poisson formula

Matching the interval

Step 1 — Identify λ\lambdaλ and kkk

Step 2 — Substitute into the formula

Step 3 — Evaluate

Test yourself

Practice by grade

Step 1 — Identify $\lambda$ and $k$

Step 1 — Identify $\lambda$ and $k$