Expected value as the long-run mean

What you will be able to do

• find the expected value of a random variable from its table;
• explain the expected value as the mean of a very long run;
• use expected values to plan, for example a total over many days.

The mean of the survey

Go back to the made-up class survey. The total number of brothers and sisters of the $40$ students is

So their mean is $\dfrac{50}{40} = 1.25$.

Now split the fraction into four parts: $0 \times \dfrac{8}{40} + 1 \times \dfrac{18}{40} + 2 \times \dfrac{10}{40} + 3 \times \dfrac{4}{40}$. Each fraction is a probability from the table. So the mean is each value times its probability, all added up.

The expected value

For any random variable, this weighted sum is called the expected value or the mean of $X$:

The Greek letter $\mu$ (mu) is another name for it. For the survey, $E(X) = 0 \times 0.2 + 1 \times 0.45 + 2 \times 0.25 + 3 \times 0.1 = 1.25$.

The expected value is not a value you expect to see. No student has $1.25$ brothers and sisters. It is the mean in the long run: pick student after student, with replacement, and the mean of their answers settles near $1.25$.

A made-up delivery van

A made-up delivery van sends back $X$ parcels a day that it could not deliver. The firm's records give this table.

So $E(X) = 0 + 0.3 + 0.3 + 0.15 = 0.75$ parcels a day. On most days the van sends back none at all, since $0$ is the most likely value. Still, over $200$ days the firm should plan for about $200 \times 0.75 = 150$ parcels.

That last step uses a useful fact. The expected value of a total is the total of the expected values. So $30$ days have an expected total of $30 \times 0.75 = 22.5$ parcels.

You should find that with $10$ days the band is wide, from about $0.19$ to about $1.31$ parcels. With $10\,000$ days it is only about $0.018$ either side of $0.75$.

Watch a simulation settle

The program below simulates $10\,000$ days of the van with Python's random module. It prints the mean of the first $10$, $100$, $1000$ and $10\,000$ days, and draws the running mean.

This is the law of large numbers, which you met in the chapter on probability rules. There it was about a share of successes. Here it is about the mean of any random variable.

Using an expected value to decide

A made-up phone shop sells a repair plan for a year. Its records say that in a year a phone needs no repair with probability $0.85$. It needs a small repair costing $2000$ rupees with probability $0.1$, and a big repair costing $8000$ rupees with probability $0.05$. The expected repair cost per phone is

If the plan costs $900$ rupees, the shop gains about $300$ rupees a plan on average over many plans. A single customer may still gain a lot. The expected value describes the long run, not one phone. The fair games of the chapter on probability rules use the same idea.

Your turn

The usual mistakes

• Taking the plain average of the values. The values are weighted by their probabilities. The plain average of $0$, $1$, $2$ and $3$ is $1.5$, not $0.75$.
• Thinking the expected value is the most likely value. The van's most likely value is $0$, but its expected value is $0.75$.
• Rounding the expected value to a whole number. $0.75$ parcels a day is right. It is a long-run mean.
• Expecting a short run to match. Ten days can give a mean far from $E(X)$.

Remember these

• $E(X) = \sum x\,P(X = x)$, the mean in the long run.
• It need not be a value that $X$ can take.
• The expected value of a total is the total of the expected values.
• A simulation of many trials settles near $E(X)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. $X$ takes the values $1$, $2$ and $3$ with probabilities $0.2$, $0.5$ and $0.3$. Find $E(X)$.

2. The answer to question 1 is not a whole number, but $X$ is a count. Is that a mistake?

3. With the $X$ of question 1, what total do you expect over $50$ independent days?

4. A table has $P(X = 0) = 0.4$, $P(X = 1) = 0.1$ and $P(X = 4) = 0.5$. Which value is the most likely, and what is $E(X)$?

5. A simulation of the van gives a mean of $1.1$ parcels a day after $8$ days. Is the program wrong?

References

• OpenStax, Introductory Statistics, Chapter 4, section on the mean or expected value and the standard deviation.