Step by step
The question
Question: X is Poisson with mean 1: find the probability that X is at most 3.
Poisson counts how many times something happens in a fixed time, when we know the average.
1 Name the numbers
μ, the average, is 1.
At most 3: x can be 0, 1, 2, 3.
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 1 is about 0.3679.
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: 1 to the power 0 is 1, and 0 factorial is 1.
1 times 0.3679, over 1, is about 0.3679.
For 1: 1 to the power 1 is 1, and 1 factorial is 1.
1 times 0.3679, over 1, is about 0.3679.
For 2: 1 to the power 2 is 1, and 2 factorial is 2.
1 times 0.3679, over 2, is about 0.1839.
For 3: 1 to the power 3 is 1, and 3 factorial is 6.
1 times 0.3679, over 6, is about 0.0613.
4 Add them
At most 3: we add them.
0.3679 plus 0.3679 plus 0.1839 plus 0.0613 is about 0.9810.
5 Check
Let's check another way: step up from P of 0.
P of 0 is e to the power minus 1: about 0.3679.
Each next one is the last one times 1, divided by the new x.
P of 1 is about 0.3679.
P of 2 is about 0.1839.
P of 3 is about 0.0613.
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.