Expected value and variance of a discrete random variable

The long-run average

The expected value of a discrete random variable is its mean — the average value of $X$ over a very long run of repetitions, each value weighted by how often it occurs:

$$\mu =E[X]=\sum xp(x)$$

It is a weighted average with the probabilities as weights. The variance is the weighted average squared distance from $\mu$, and the shortcut form is the one to compute by hand:

$${\sigma }^2=\mathrm{Var}(X)=\sum (x-\mu {)}^{2}p(x)=\sum x^{2}p(x)-{\mu }^2$$

The standard deviation is $\sigma =\sqrt{{\sigma }^2}$, in the units of $X$.

Worked example — the table method

Extend Maple Ledger's pmf with two columns and total each:

$$\mu =0.95,E[X^2]=1.85,{\sigma }^2=1.85-0.95^2=1.85-0.9025=0.9475,\sigma =0.973$$

On average a batch has 0.95 late invoices, give or take about one. No batch ever has 0.95 late invoices — the expected value is a long-run average, not a possible outcome, exactly as a family can have 1.8 children. The third column is $x^{2}p(x)$, not $(xp(x){)}^2$: square the value, then weight it.

Forgetting the $-{\mu }^2$

The most common slip is to report the third column's total, 1.85, as the variance. That is $E[X^2]$; the variance is smaller by ${\mu }^2$. A variance larger than $E[X^2]$ is impossible, so a quick check is that ${\sigma }^2<\sum x^{2}p(x)$ always.

Expected value as a decision rule

Maple Ledger's partner can install a review control for \$8,000 a year, or accept that without it there is a 12 % chance of a \$50,000 loss from an undetected billing error. The expected loss without the control is

$$E[\text{loss}]=0.12\times 50,000+0.88\times 0=\$6,000$$

which is less than \$8,000 — on expected value alone, skip the control. But expected value is an average over many years; in any single year the firm either loses nothing or loses \$50,000, and a firm that cannot absorb \$50,000 may rationally pay \$8,000 to avoid the chance. Expected value ranks decisions by their long-run average; the risk appetite — how much variance the decision-maker will carry — is a separate input the paper often asks you to name.