 
Summary: Independent families of random variables.
Definition. Suppose X is a family of random variables. We say X is independent if
P(X1 A1, . . . , Xn An) = P(X1 A1) · · · P(Xn An)
whenever X1, . . . , Xn are distinct members of X and A1, . . . , An are Borel subsets of R.
Proposition. Suppose X is a family of random variables. Then X is independent if and only if
FX1,...,Xn (x1, . . . , xn) = FX1 (x1) · · · FXn (xn)
whenever X1, . . . , Xn are distinct members of X and (x1, . . . , xn) Rn
.
Proof. This is really simple once you get straight what a Borel set is. We won't do this, though.
Proposition. Suppose X1, . . . , Xn are distinct discrete random variables. Then {X1, . . . , Xn} is indepen
dent if and only if
(1) pX1,...,Xn (x1, . . . , xn) = pX1 (x1) · · · pXn (xn) whenever (x1, . . . , xn) Rn
.
Proof. This is long winded but simple minded. I hope you will see this.
Suppose {X1, . . . , Xn} is independent and (x1, . . . , xn) Rn
. Let Ai = {xi} for each i = 1, . . . , n. Then
pX1,...,Xn (x1, . . . , xn) = P(X1 = x1, . . . , Xn = xn)
= P(X1 A1, . . . , Xn An)
= P(X1 A1) · · · P(Xn An)
= P(X1 = x1) · · · P(Xn = xn)
