x: 2, 4, 6, 8; P(x): 0.1, 0.4, 0.3, 0.2, find the expected value Step by Step

Step by step

The question

Question: x can be 2, 4, 6, 8, with probabilities 0.1, 0.4, 0.3, 0.2: find the expected value.

x: 2, 4, 6, 8; P(x): 0.1, 0.4, 0.3, 0.2. expected value?

The expected value is the mean of x: what we get on average, in the long run.

expected value = the mean of x, μ

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.

0.1 plus 0.4 plus 0.3 plus 0.2 is 1.

Good.

0.1 + 0.4 + 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 0.1 is 0.2.

4 times 0.4 is 1.6.

6 times 0.3 is 1.8.

8 times 0.2 is 1.6.

3 Add the column

0.2 plus 1.6 plus 1.8 plus 1.6 is 5.2.

The mean, μ, is 5.2.

μ = 0.2 + 1.6 + 1.8 + 1.6 = 5.2

Careful!

The plain average treats every x the same.

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

Careful: not (2 + 4 + 6 + 8) ÷ 4 = 5 ✗

4 Check

Let's check with 10 tries.

Each x comes up about 10 times its probability: 2, 1 times; 4, 4 times; 6, 3 times; 8, 2 times.

10 tries: 2 comes 1 times, 4 comes 4 times, 6 comes 3 times, 8 comes 2 times

All the x's of the 10 tries add up to 52.

total 2 × 1 + 4 × 4 + 6 × 3 + 8 × 2 = 52

52 divided by 10 is 5.2: the mean.

So it is right!

52 ÷ 10 = 5.2

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer