Poisson mean 3, P(X >= 2) Step by Step | The Poisson distribution

Step by step

The question

Question: X is Poisson with mean 3: find the probability that X is at least 2.

Poisson, μ = 3: P(X ≥ 2)?

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

At least 2: x can be 2, 3, 4, and so on, with no end.

x = 2, 3, 4, ...

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 Use the rest

x can go on with no end, so we cannot add them all.

All the probabilities add up to 1: so we take away the ones below 2.

P(X ≥ 2) = 1 − P(X < 2)

For 0: 3 to the power 0 is 1, and 0 factorial is 1.

1 times 0.0498, over 1, is about 0.0498.

P(X = 0) = 3⁰ × e⁻³0! = 1 × 0.04981 ≈ 0.0498

For 1: 3 to the power 1 is 3, and 1 factorial is 1.

3 times 0.0498, over 1, is about 0.1494.

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

1 minus the sum 0.0498 plus 0.1494 is about 0.8009.

P(X ≥ 2) = 1 − (0.0498 + 0.1494) ≈ 0.8009

4 Check

Let's check the key number another way: e to the power 3 is about 20.0855.

1 divided by it is about 0.0498: the same as e to the power minus 3.

e³ ≈ 20.0855, 1 ÷ 20.0855 ≈ 0.0498

And step up: each next P is the last one times 3, divided by the new x.

P of 1 is about 0.1494.

The same as above.

So it is right!

P(1) = P(0) × 3 ÷ 1 ≈ 0.1494

Let's look back.

First: name the numbers.

Then: write the formula.

Then: use the rest.

Last: check.

Answer