Expected Value and Standard Deviation

The expected value is simply the long-run average of a random variable: what you'd get per try, on average, over thousands of tries. The formula $\sum x\,P(x)$ is short: each value $x$, times $P(x)$ (how often it comes up), all added together. The variance formula has more symbols but the same structure: $(x - \mu)^2$ (how far each value is from the average, squared), weighted by $P(x)$ and added up. It's exactly the standard deviation from descriptive statistics, with probabilities playing the role of "how many times each value appeared."

Why weight by probability? Think about rolling a die 600 times. You'd expect about 100 of each face, so the total is about $100(1 + 2 + \dots + 6)$, and the average per roll is that total over 600. Dividing each 100 by 600 gives $\frac16$, the probability. So $\sum x\,P(x)$ is just the ordinary average of a very long run, written with probabilities instead of counts.

Expected value as a balance point. Draw the distribution as bars: $\mu$ is where it would balance on a fulcrum, just like the mean of a data set.

Fair games have expected net gain 0. Casinos and insurers choose prices so that your expected gain is negative and theirs is positive; the law of large numbers then guarantees their profit over many bets.

A common mistake. The word "expected" makes it tempting to think it's the value you should expect to see. It needn't even be possible. A die's expected value is $3.5$, though no roll gives $3.5$. It's a long-run average, not a prediction for one try.

Insurers set premiums from expected payouts, investors weigh expected returns against standard deviation (risk), and AI agents choose actions that maximize expected reward.