Step by step
The question
Question: X is Poisson with mean 3: find the probability that X is at most 2.
Poisson counts how many times something happens in a fixed time, when we know the average.
1 Name the numbers
μ, the average, is 3.
At most 2: x can be 0, 1, 2.
2 Write the formula
The Poisson formula: μ to the power x, times e to the power minus μ, over x factorial.
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.
Careful!
0 factorial is 1, not 0.
3 One P for each x
For 0: 3 to the power 0 is 1, and 0 factorial is 1.
1 times 0.0498, over 1, is about 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.
For 2: 3 to the power 2 is 9, and 2 factorial is 2.
9 times 0.0498, over 2, is about 0.224.
4 Add them
At most 2: we add them.
0.0498 plus 0.1494 plus 0.2240 is about 0.4232.
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.
Each next one is the last one times 3, divided by the new x.
P of 1 is about 0.1494.
P of 2 is about 0.2240.
The same numbers as above.
So it is right!
Let's look back.
First: name the numbers.
Then: write the formula.
Then: one P for each x.
Then: add them.
Last: check.