Step by step
The question
Question: x can be 10, 20, 30, with probabilities 0.2, 0.5, 0.3: find the mean.
1 Check the table
The table lists each value of x and its probability, P of x.
A probability table must add up to 1.
0.2 plus 0.5 plus 0.3 is 1.
Good.
2 Multiply x × P(x)
The mean of a random variable: each x times its probability, all added up.
10 times 0.2 is 2.
20 times 0.5 is 10.
30 times 0.3 is 9.
3 Add the column
2 plus 10 plus 9 is 21.
The mean, μ, is 21.
Careful!
The plain average treats every x the same.
But the likely ones count more: we weight by P of x.
4 Check
Let's check with 10 tries.
Each x comes up about 10 times its probability: 10, 2 times; 20, 5 times; 30, 3 times.
All the x's of the 10 tries add up to 210.
210 divided by 10 is 21: the mean.
So it is right!
Let's look back.
First: check the table.
Then: multiply x × P(x).
Then: add the column.
Last: check.