Basic Probability

Abstract

Abstract A probability space is a measure space (n, :F, P) such that P(O) = 1. A complete probability space is a probability space such that each subset of a P-null set is measurable. We never assume that a probability space is complete unless we so state. In the context of probability, measurable sets are called events. A random variable is a measurable real-valued function defined on a probability space. A random process is an indexed family of random variables, each of which is defined on the same probability space. In this chapter, for a random process {X(t, w): t ER} defined on a probability space (0, :F, P), we will let X(t) denote X(t, ). Example 7.1. A nonnegative Riemann integrable .function or a nonnegative Lebesgue integrable .function that integrates to one need not be a probability density function. Proof: Let (0, :F, P) be a probability space and let X be a random variable defined on (0, :F, P). For each Borel subset B of R, let μ(B) = P(X EB). Note that µis a measure mapping the real Borel sets into (0, 1). This