ABSTRACT
A real-valued point function X(·) defined on the probability space (Ω, F, P) is called a “real random variable” (or sometimes just a “random variable”) if the set {ω ∈ Ω : X(ω) ≤ x] ∈ F for every real number x. We would like to be able to assign a probability to the event “X is less than or equal to x” and so must require that this set be in the σ-field of measurable events. In fact, the point function F(x) = P{ω ∈ Ω : X(ω) ≤ x} = P(X ≤ x) is called the “cumulative distribution function” of the random variable X. Knowledge of the cumulative distribution function of a random variable allows us to compute probabilities for any Borel-measurable set E, other than just those of the form {ω ∈ Ω: X(ω) ≤ x} for some real x, using the relationship:
