x: -2, 0, 2, 4; P(x): 0.2, 0.3, 0.3, 0.2, find the mean Step by Step

Step by step

The question

Question: x can be −2, 0, 2, 4, with probabilities 0.2, 0.3, 0.3, 0.2: find the mean.

x: −2, 0, 2, 4; P(x): 0.2, 0.3, 0.3, 0.2. 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 0.3 plus 0.3 plus 0.2 is 1.

Good.

0.2 + 0.3 + 0.3 + 0.2 = 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)

−2 times 0.2 is −0.4.

0 times 0.3 is 0.

2 times 0.3 is 0.6.

4 times 0.2 is 0.8.

3 Add the column

−0.4 plus 0 plus 0.6 plus 0.8 is 1.

The mean, μ, is 1.

μ = (−0.4) + 0 + 0.6 + 0.8 = 1

Careful!

Each x counts as much as its probability.

Careful: each x counts as much as its P(x)

4 Check

Let's check with 10 tries.

Each x comes up about 10 times its probability: −2, 2 times; 0, 3 times; 2, 3 times; 4, 2 times.

10 tries: −2 comes 2 times, 0 comes 3 times, 2 comes 3 times, 4 comes 2 times

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

total −2 × 2 + 0 × 3 + 2 × 3 + 4 × 2 = 10

10 divided by 10 is 1: the mean.

So it is right!

10 ÷ 10 = 1

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer