x: 0, 1, 2, 3; P(x): 0.15, 0.35, 0.3, 0.2, find the mean Step by Step

Step by step

The question

Question: x can be 0, 1, 2, 3, with probabilities 0.15, 0.35, 0.3, 0.2: find the mean.

x: 0, 1, 2, 3; P(x): 0.15, 0.35, 0.3, 0.2. 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.

0.15 plus 0.35 plus 0.3 plus 0.2 is 1.

Good.

0.15 + 0.35 + 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)

0 times 0.15 is 0.

1 times 0.35 is 0.35.

2 times 0.3 is 0.6.

3 times 0.2 is 0.6.

3 Add the column

0 plus 0.35 plus 0.6 plus 0.6 is 1.55.

The mean, μ, is 1.55.

μ = 0 + 0.35 + 0.6 + 0.6 = 1.55

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 = 1.5 ✗

4 Check

Let's check with 100 tries.

Each x comes up about 100 times its probability: 0, 15 times; 1, 35 times; 2, 30 times; 3, 20 times.

100 tries: 0 comes 15 times, 1 comes 35 times, 2 comes 30 times, 3 comes 20 times

All the x's of the 100 tries add up to 155.

total 0 × 15 + 1 × 35 + 2 × 30 + 3 × 20 = 155

155 divided by 100 is 1.55: the mean.

So it is right!

155 ÷ 100 = 1.55

Let's look back.

First: check the table.

Then: multiply x × P(x).

Then: add the column.

Last: check.

Answer