Expected value represents the probability-weighted average of all possible values that a random variable can produce.
It serves as the primary measure of central tendency in theoretical probability models.
For a discrete random variable, it is computed by multiplying each possible value by its probability and summing the results.
For a continuous random variable, it is calculated by integrating the product of the value and its probability density function.
In mathematical notation, expected value is denoted by capital E followed by the random variable name in brackets.
A fundamental property of expected value is linearity: the expectation of a sum of variables equals the sum of their individual expectations.
Linearity of expectation holds true universally, regardless of whether the variables are statistically independent or correlated.
This powerful property allows data scientists to compute expectations for complex aggregated systems by decomposing them into simple parts.
The expected value of a game or investment does not guarantee that any single trial will land near that average figure.
For example, a lottery ticket with an expected return of negative two pounds produces zero or millions on individual plays.
The Petersburg paradox demonstrates that expected value alone is insufficient for decision making when variance is infinite.
Expected value assumes risk neutrality, whereas human decision makers and businesses typically exhibit risk aversion.
Expected utility theory extends expected value by transforming raw financial payouts into subjective utility scores before averaging.
Some heavy-tailed probability distributions, such as the Cauchy distribution, have no defined expected value because integrals diverge.
Analysts must confirm that an expectation exists mathematically before relying on sample averages to estimate it.