Elements of Statistical Methods Probability (Ch 3) - Statistics

Induced Events and ProbabilitiesSuppose we have a r.v. X : S −→ R with corresponding probability space (S,C,P).For any Borel set B ∈ B we can determine the set X −1 (B) of all outcomes in Swhich get mapped into B, i.e.,X −1 (B) = {s ∈ S : X(s) ∈ B}How do we know that X −1 (B) ∈ C is an event?We don’t.Thus we require it in our definition of a random variable.Definition: A function X : S −→ R is a random variable if and only if theinduced event X −1 ((−∞,y]) ∈ C for any y ∈ Rand thus the induced probability P X (induced by P and X)P X ((−∞,y]) = P({s ∈ S : X(s) ≤ y})exists for all y ∈ R40