Discrete & Continuous Random Variables - Statistics

ExpectationThis fair value of the game is formalized in the notion of the expected valueor expectation of the game pay-off X.Definition: For a discrete random variable X the expected value E(X), or simplyEX, is defined as the probability weighted average of the possible values of X,i.e.,EX =∑ x · P(X = x) =x∈X(S)∑ x · f (x) = ∑x · f (x)x∈X(S)xThe expected value of X is also called the population mean and denoted by µ.Example (Bernoulli r.v.): If X ∼ Bernoulli(p) thenµ = EX = ∑ x · P(X = x) = 0 · P(X = 0) + 1 · P(X = 1) = P(X = 1) = px∈{0,1}14