x: 0, 1, 2, 3, 4; P(x): 0.05, 0.15, 0.4, 0.3, 0.1, find the mean Step by Step

Step by step

The question

Question: x can be 0, 1, 2, 3, 4, with probabilities 0.05, 0.15, 0.4, 0.3, 0.1: find the mean.

x: 0, 1, 2, 3, 4; P(x): 0.05, 0.15, 0.4, 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.

0.05 plus 0.15 plus 0.4 plus 0.3 plus 0.1 is 1.

Good.

0.05 + 0.15 + 0.4 + 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)

0 times 0.05 is 0.

1 times 0.15 is 0.15.

2 times 0.4 is 0.8.

3 times 0.3 is 0.9.

4 times 0.1 is 0.4.

3 Add the column

0 plus 0.15 plus 0.8 plus 0.9 plus 0.4 is 2.25.

The mean, μ, is 2.25.

μ = 0 + 0.15 + 0.8 + 0.9 + 0.4 = 2.25

Careful!

The plain average treats every x the same.

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

Careful: not (0 + 1 + 2 + 3 + 4) ÷ 5 = 2 ✗

4 Check

Let's check with 100 tries.

Each x comes up about 100 times its probability: 0, 5 times; 1, 15 times; 2, 40 times; 3, 30 times; 4, 10 times.

100 tries: 0 comes 5 times, 1 comes 15 times, 2 comes 40 times, 3 comes 30 times, 4 comes 10 times

All the x's of the 100 tries add up to 225.

total 0 × 5 + 1 × 15 + 2 × 40 + 3 × 30 + 4 × 10 = 225

225 divided by 100 is 2.25: the mean.

So it is right!

225 ÷ 100 = 2.25

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer