Conditional Probability and Expectation

Abstract

This chapter discusses conditional probability and expectation. Conditional probabilities occur naturally in many problems. It presents an assumption that in a random variable X, where X has distribution function F, if X takes the value x, a random variable Y is observed, where the distribution of Y depends on x. Thus P(x, B) = P{Y ∈ BI X = x} is prescribed in the statement of the problem, although the event {X = x} may have probability zero for all values of x. Given X = x, (X, Y) will lie in C iff Y belongs to the section C(x) ={y: (x, y) ∈ C}. The probability of this event is P(x, C(x)). If P(x, B) is Borel measurable in x for each fixed B ∈ (R), then by the product measure theorem, there is a unique probability measure on (R2). Thus, in the mathematical formulation of the problem, the probability measure P on = (R2) is taken to be the unique measure determined by PX and the measures P(x, ·), x ∈ R.