x: 0, 2, 4; P(x): 0.3, 0.4, 0.3, find the expected value Step by Step

Step by step

The question

Question: x can be 0, 2, 4, with probabilities 0.3, 0.4, 0.3: find the expected value.

x: 0, 2, 4; P(x): 0.3, 0.4, 0.3. 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.

0.3 plus 0.4 plus 0.3 is 1.

Good.

0.3 + 0.4 + 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 0.3 is 0.

2 times 0.4 is 0.8.

4 times 0.3 is 1.2.

3 Add the column

0 plus 0.8 plus 1.2 is 2.

The mean, μ, is 2.

μ = 0 + 0.8 + 1.2 = 2

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: 0, 3 times; 2, 4 times; 4, 3 times.

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

All the x's of the 10 tries add up to 20.

total 0 × 3 + 2 × 4 + 4 × 3 = 20

20 divided by 10 is 2: the mean.

So it is right!

20 ÷ 10 = 2

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer