x: 1, 2, 3, 4, 5; P(x): 0.1, 0.2, 0.3, 0.3, 0.1, find the expected… Step by Step

Step by step

The question

Question: x can be 1, 2, 3, 4, 5, with probabilities 0.1, 0.2, 0.3, 0.3, 0.1: find the expected value.

x: 1, 2, 3, 4, 5; P(x): 0.1, 0.2, 0.3, 0.3, 0.1. 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 0.2 plus 0.3 plus 0.3 plus 0.1 is 1.

Good.

0.1 + 0.2 + 0.3 + 0.3 + 0.1 = 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)

1 times 0.1 is 0.1.

2 times 0.2 is 0.4.

3 times 0.3 is 0.9.

4 times 0.3 is 1.2.

5 times 0.1 is 0.5.

3 Add the column

0.1 plus 0.4 plus 0.9 plus 1.2 plus 0.5 is 3.1.

The mean, μ, is 3.1.

μ = 0.1 + 0.4 + 0.9 + 1.2 + 0.5 = 3.1

Careful!

The plain average treats every x the same.

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

Careful: not (1 + 2 + 3 + 4 + 5) ÷ 5 = 3 ✗

4 Check

Let's check with 10 tries.

Each x comes up about 10 times its probability: 1, 1 times; 2, 2 times; 3, 3 times; 4, 3 times; 5, 1 times.

10 tries: 1 comes 1 times, 2 comes 2 times, 3 comes 3 times, 4 comes 3 times, 5 comes 1 times

All the x's of the 10 tries add up to 31.

total 1 × 1 + 2 × 2 + 3 × 3 + 4 × 3 + 5 × 1 = 31

31 divided by 10 is 3.1: the mean.

So it is right!

31 ÷ 10 = 3.1

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer