x: 0, 1, 2; P(x): 0.2, 0.5, 0.3, find the mean Step by Step

Step by step

The question

Question: x can be 0, 1, 2, with probabilities 0.2, 0.5, 0.3: find the mean.

x: 0, 1, 2; P(x): 0.2, 0.5, 0.3. 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.

0.2 + 0.5 + 0.3 = 1

2 Multiply x × P(x)

The mean of a random variable: each x times its probability, all added up.

mean μ = add up all x × P(x)

0 times 0.2 is 0.

1 times 0.5 is 0.5.

2 times 0.3 is 0.6.

3 Add the column

0 plus 0.5 plus 0.6 is 1.1.

The mean, μ, is 1.1.

μ = 0 + 0.5 + 0.6 = 1.1

Careful!

The plain average treats every x the same.

But the likely ones count more: we weight by P of x.

Careful: not (0 + 1 + 2) ÷ 3 = 1 ✗

4 Check

Let's check with 10 tries.

Each x comes up about 10 times its probability: 0, 2 times; 1, 5 times; 2, 3 times.

10 tries: 0 comes 2 times, 1 comes 5 times, 2 comes 3 times

All the x's of the 10 tries add up to 11.

total 0 × 2 + 1 × 5 + 2 × 3 = 11

11 divided by 10 is 1.1: the mean.

So it is right!

11 ÷ 10 = 1.1

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer