x: 1, 2, 3, 4, 5, 6; P(x): 0.1, 0.1, 0.2, 0.2, 0.3, 0.1, find the mean Step by Step

Step by step

The question

Question: x can be 1, 2, 3, 4, 5, 6, with probabilities 0.1, 0.1, 0.2, 0.2, 0.3, 0.1: find the mean.

x: 1, 2, 3, 4, 5, 6; P(x): 0.1, 0.1, 0.2, 0.2, 0.3, 0.1. 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.

0.1 plus 0.1 plus 0.2 plus 0.2 plus 0.3 plus 0.1 is 1.

Good.

0.1 + 0.1 + 0.2 + 0.2 + 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 0.1 is 0.2.

3 times 0.2 is 0.6.

4 times 0.2 is 0.8.

5 times 0.3 is 1.5.

6 times 0.1 is 0.6.

3 Add the column

0.1 plus 0.2 plus 0.6 plus 0.8 plus 1.5 plus 0.6 is 3.8.

The mean, μ, is 3.8.

μ = 0.1 + 0.2 + 0.6 + 0.8 + 1.5 + 0.6 = 3.8

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 + 6) ÷ 6 = 3.5 ✗

4 Check

Let's check with 10 tries.

Each x comes up about 10 times its probability: 1, 1 times; 2, 1 times; 3, 2 times; 4, 2 times; 5, 3 times; 6, 1 times.

10 tries: 1 comes 1 times, 2 comes 1 times, 3 comes 2 times, 4 comes 2 times, 5 comes 3 times, 6 comes 1 times

All the x's of the 10 tries add up to 38.

total 1 × 1 + 2 × 1 + 3 × 2 + 4 × 2 + 5 × 3 + 6 × 1 = 38

38 divided by 10 is 3.8: the mean.

So it is right!

38 ÷ 10 = 3.8

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer