General Expectation So far we have considered expected values separately for discrete and continuous random variables only. This separation is somewhat unnatural. Further, some random variables are neither discrete nor continuous – for example mixture of discrete and continuous distributions. Therefore, we need a more general definition of expected value. STA347 - week 7

Definition Let X be an arbitrary random variable (perhaps neither discrete nor continuous). Then, the expected value of X is given by STA347 - week 7

Theorem 1 Let X be a discrete random variable with distinct possible values x1, x2,…, and let pi = P(X = xi). Then, Proof: STA347 - week 7

Theorem 2 Let X be continuous random variable with density fX. Then,   Proof: STA347 - week 7

Application to Mixture Distribution Let Yi be a random variable with cdf Fi for 1≤ i ≤ k. Let X be a random variable whose cdf corresponds to a finite mixture of the cdfs of the Yi, so that , where pi ≥ 0 and .Then, Proof: STA347 - week 7