Elements of Statistical Methods Probability (Ch 3) - Statistics

Two Independent Random VariablesTwo random variable X 1 and X 2 are independent if any events defined in terms ofX 1 is independent of any event defined in terms of X 2 .The following weaker but more practical definition is equivalent.Definition: Let X 1 : S −→ R and X 2 : S −→ R be random variables defined on thesame sample space S. X 1 and X 2 are independent if and only if for each y 1 ∈ Rand y 2 ∈ RP(X 1 ≤ y 1 , X 2 ≤ y 2 ) = P(X 1 ≤ y 1 ) · P(X 2 ≤ y 2 )Note the shorthand notationP(X 1 ≤ y 1 , X 2 ≤ y 2 ) = P({X 1 ≤ y 1 } ∩ {X 2 ≤ y 2 })You also often see P(AB) for P(A ∩ B).47