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Lecture 14: Cumulative Distribution Functions and Continuous Random Variables 1. Properties of the CDF Proposition: Let X be a real-valued random variable (not necessarily discrete) with cumulative distribution function (CDF) F (x) = P(X ≤ x). Then a) F is non-decreasing, i.e., if x < y, then F (x) ≤ F (y). b) limx→∞ F (x) = 1. c) limx→−∞ F (x) = 0. d) F is right-continuous, i.e., for any x and any decreasing sequence (xn , n ≥ 1), that converges to x, limn→∞ F (xn ) = F (x). Proof: a) If x < y, then {X ≤ x} ⊂ {X ≤ y}, which implies that F (x) ≤ F (y). b) If (xn , n ≥ 1) is an increasing sequence such that xn → ∞, then the events En = {X ≤ xn } form an increasing sequence with {X < ∞} = ∞ [ En . n=1 It follows from the continuity properties of probability measures (Lecture 2) that lim F (xn ) = lim P(En ) = P(X < ∞) = 1. n→∞ n→∞ c) Likewise, if (xn , n ≥ 1) is a decreasing sequence such that xn → −∞, then the events En = {X ≤ xn } form a decreasing sequence with ∅= ∞ \ En . n=1 In this case, the continuity properties of measures imply that lim F (xn ) = lim P(En ) = P(∅) = 0. n→∞ n→∞ d) If xn , n ≥ 1 is a decreasing sequence converging to x, then the sets En = {X ≤ xn } also form a decreasing sequence with ∞ \ {X ≤ x} = En . n=1 Consequently, lim F (xn ) = lim P(En ) = P{X ≤ x} = F (x). n→∞ n→∞ Some other notable properties of the CDF are: 1 i) P(a < X ≤ b) = F (b) − F (a) for all a < b. ii) Another application of the continuity property shows that the probability that X is strictly less than x is equal to P(X < x) = lim P(X ≤ x − 1/n) = n→∞ lim F (x − 1/n). n→∞ In other words, P(X < x) is equal to the left limit of F at x, which is often denoted F (x−). However, notice that in general this limit need not be equal to F (x): F (x) = P(X < x) + P(X = x), and so P(X < x) = F (x) if and only if P(X = x) = 0. iii) The converse to this proposition is also true: namely, if F : R → [0, 1] is a function satisfying properties (a) - (d), then F is the CDF of some random variable X. 2.) Continuous Random Variables Definition: A real-valued random variable X is said to be a continuous random variable if there is a non-negative function f : R → [0, ∞) such that Z b P(a ≤ X ≤ b) = f (x)dx a for all a < b. The function f is called the probability density function of X. Remarks: Notice that Z ∞ 1 = P(−∞ < X < ∞) = f (x)dx, −∞ and so the integral of a density function over the entire real line must be equal to 1. Since Z P(X = x) = x f (y)dy = 0, x the probability that a continuous RV will assume any particular value is 0. There is a close connection between the density of a random variable and its CDF. First, observe that Z x F (x) = P(X ≤ x) = f (y)dy. ∞ Since the right-hand side is a differentiable function of x, it follows that F (x) is also differentiable and that F 0 (x) = f (x), 2 i.e., the density is the derivative of the cumulative distribution function. Example: If X is a continuous RV with CDF Fx and density fx , find the CDF and the density function of Y = 2X. Solution: The CDF of Y is equal to FY (x) = P(Y ≤ x) = P(2X ≤ x) = P(X ≤ x/2) = FX (x/2), and differentiation with respect to x gives fY (x) = 1 d FY (x) = fX (x/2). dx 2 3