Solve P(X = 1), Poisson mean 3 Step by Step

Step by step

The question

Question: X is Poisson with mean 3: find the probability that X is exactly 1.

Poisson, μ = 3: P(X = 1)?

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 3.

μ = 3

x, the count we want, is 1.

x = 1

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 3 is about 0.0498.

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

e ≈ 2.718, e⁻³ ≈ 0.0498

x! = x × ... × 2 × 1, 0! = 1

Careful!

0 factorial is 1, not 0.

Careful: 0! = 1 (not 0)

3 Put the numbers in

3 to the power 1 is 3.

1 factorial is 1: 1.

3¹ = 3, 1! = 1 = 1

4 Work it out

3 times 0.0498, over 1, is about 0.1494.

P(X = 1) = 3¹ × e⁻³1! = 3 × 0.04981 ≈ 0.1494

5 Check

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

P of 0 is e to the power minus 3: about 0.0498.

P(0) = e⁻³ ≈ 0.0498

Each next one is the last one times 3, divided by the new x.

P of 1 is about 0.1494.

P(1) = P(0) × 3 ÷ 1 ≈ 0.1494

Let's look back.

First: name the numbers.

Then: write the formula.

Then: put the numbers in.

Then: work it out.

Last: check.

Answer