The law of a random variable: “For a random variable X, the event {ω | X(ω) ≤ c} is often written as {X ≤ c}, and is sometimes just called “the event that X ≤ c.” The probability of this event is well defined, since this event belongs to F. Let now B be a more general
subset of the real line. We use the notation X−1(B) or {X ∈B} to denote the set {ω | X(ω) ∈B}”
Proof: Clearly, PX(B) ≥ 0, for every Borel set B. Also, PX(R) = P(X ∈R) = P(Ω) = 1. We now verify countable additivity. Let {Bi} be a countable sequence of disjoint Borel subsets of R. Note that the sets X−1(Bi) are also disjoint, and that
Discrete random variables:
Discrete random variables take values in a countable set. We need some notation. Given a function f : Ω → R, its range is the set f(Ω) = {x ∈R | ∃ω∈Ωsuch that f(ω) = x}. Discrete random variables and PMFs)
(a)A random variable X, defined on a probability space (Ω, F, P), is said to be discrete if its range X(Ω) is countable.
(b) If X is a discrete random variable, the function pX : R → [0, 1] defined by pX(x) = P(X = x), for every x, is called the (probability) mass function of X, or PMF for short. Consider a discrete random variable X whose range is a finite set C. In that case, for any Borel set A, countable additivity yields