Expected Value

Expected value is the probability-weighted long-run average of a random variable.

A probability distribution pairs each outcome with a nonnegative weight totalling one. Discrete outcomes have probability bars whose heights sum to one. A weighted balance marker locates expected value, which need not itself be a possible outcome.

Expected value balances every possible outcome by how often the model says it should occur in the long run.

Definition

For a discrete random variable $X$ with possible values $x_i$,

E[X]=\mu=\sum_i x_iP(X=x_i).

Multiply each value by its probability, then add. The probabilities must represent a complete valid distribution.

Long-run interpretation

If the random process is repeated independently many times under stable conditions, the average outcome tends to approach $E[X]$.

Expected value need not be a possible single outcome. The expected number on a fair die is $3.5$, even though no roll shows $3.5$.

Worked example: a simple game

This does not mean each play earns ten cents; it describes the long-run average.

Insurance interpretation

Insurance pools many uncertain losses. An expected claim cost helps set premiums, but administrative costs, capital requirements, risk, and profit also matter.

An individual may rationally pay more than expected loss to reduce exposure to a rare severe outcome.

Common mistakes

Averaging possible values without probability weights. Outcomes may not be equally likely.

Using prize instead of net gain. Subtract any cost consistently.

Interpreting expectation as the next outcome. It is a long-run mean.

Ignoring outcomes with zero or negative values. Include the complete distribution.

Choosing solely by expectation when risk matters. Compare variability and consequences too.

Quick self-check

Are all possible values and probabilities included?
Do the probabilities sum to $1$?
Are losses represented with negative signs and costs included?
Is the result interpreted as a long-run average?
Does a binomial setting permit the shortcut $np$?
Should variance, feasibility, or risk also influence the decision?