Poisson mean 5, P(X = 3) Step by Step | The Poisson distribution

Step by step

The question

Question: X is Poisson with mean 5: find the probability that X is exactly 3.

Poisson, μ = 5: P(X = 3)?

Poisson counts how many times something happens in a fixed time, when we know the average.

Poisson: counts in a time or a space, with a known average

1 Name the numbers

μ, the average, is 5.

μ = 5

x, the count we want, is 3.

x = 3

2 Write the formula

The Poisson formula: μ to the power x, times e to the power minus μ, over x factorial.

P(X = x) = μ^x × e^−μx!

e is a number, about 2.718.

e to the power minus 5 is about 0.0067.

x factorial multiplies down to 1, and 0 factorial is 1.

e ≈ 2.718, e⁻⁵ ≈ 0.0067

x! = x × ... × 2 × 1, 0! = 1

Careful!

0 factorial is 1, not 0.

Careful: 0! = 1 (not 0)

3 Put the numbers in

5 to the power 3 is 125.

3 factorial is 3 times 2 times 1: 6.

5³ = 125, 3! = 3 × 2 × 1 = 6

4 Work it out

125 times 0.0067, over 6, is about 0.1404.

P(X = 3) = 5³ × e⁻⁵3! = 125 × 0.00676 ≈ 0.1404

5 Check

Let's check another way: step up from P of 0.

P of 0 is e to the power minus 5: about 0.0067.

P(0) = e⁻⁵ ≈ 0.0067

Each next one is the last one times 5, divided by the new x.

P of 1 is about 0.0337.

P(1) = P(0) × 5 ÷ 1 ≈ 0.0337

P of 2 is about 0.0842.

P(2) = P(1) × 5 ÷ 2 ≈ 0.0842

P of 3 is about 0.1404.

The same!

So it is right!

P(3) = P(2) × 5 ÷ 3 ≈ 0.1404

Let's look back.

First: name the numbers.

Then: write the formula.

Then: put the numbers in.

Then: work it out.

Last: check.

Answer