Elements of Statistical Methods Probability (Ch 3) - Statistics

General CDF Properties1. 0 ≤ F(y) ≤ 1 for all y ∈ R (F(y) is a probability)2. y 1 ≤ y 2 =⇒ F(y 1 ) ≤ F(y 2 ) (monotonicity property)This follows since {X ≤ y 1 } ⊂ {X ≤ y 2 } =⇒ P(X ≤ y 1 ) ≤ P(X ≤ y 2 ).3. Limiting behavior as we approach ±∞lim F(y) = 0 and limy→−∞ y→∞ F(y) = 1This follows (with some more attention to technical detail) since\[lim {X ≤ y} = {X ≤ y} = /0 and lim {X ≤ y} = {X ≤ y} = Sy→−∞ y→∞y→−∞y→∞Note that in our examples we had F(y) = 0 for sufficiently low y (y < 0) andF(y) = 1 for sufficiently high y. X had a finite and thus bounded value set.46