Expected Value

Probability · Axiom Academy

The long-run average value of a random variable 1. Definition of Expected Value The expected value (or expectation) of a discrete random variable X is the weighted average of all possible values, weighted by their probabilities. We sum over all possible values x, multiplying each value by its probability. The symbol μ (mu) is often used to denote E[X]. 2. Interpretation as Weighted Average Expected value is NOT necessarily a value the random variable can actually take. It's the long-run average or "center of gravity" of the distribution. Example: For a fair die, E[X] = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5, even though the die can never show 3.5. Values with higher probability contribute more to the expected value, while rare values contribute less. Expected value has a crucial property called linearity : Property 1: E[aX + b] = aE[X] + b for constants a and b Property 2: E[X + Y] = E[X] + E[Y] (this holds even if X and Y are dependent!) These properties make expected value calculations much easier in complex scenarios.

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