Download Several Random Variables

1.4
Several Random Variables
Many applications of probability theory involve two or more random variables. In this
case some of the definitions in the previous sections are extended to this case.
Definition 1. Random variables T and R are independent  {T  E} and {R  F} are
independent for any Borel sets E and F, i.e.
P(T  E and R  F) = P(T  E) P(R  F)
Random variables T1, ..., Tn are independent  {T1  E1}, …, {Tn  En} are
independent for any Borel sets E1, ..., En, i.e.
Pr{T1  E1, ..., Tn  En} = Pr{T1  E1} ... Pr{Tn  En}
Proposition 1. T and R are independent  {T  t} and {R  s} are independent for any
t and s.
Proof.  follows from the definition of independence and the fact that (- , t] and
(- , s] are Borel sets in R. To show  we must show that
P(T  t and R  s) = P(T  t)P(R  s) for all s and t implies
P(T  E and R  F) = P(T  E)P(R  F) for all Borel sets E and F. Therefore, suppose
P(T  t and R  s) = P(T  t)P(R  s) for all s and t. First we fix s and show
P(T  E and R  s) = P(T  E)P(R  s) for all Borel sets E. To do this consider the two
set functions (E) = P(T  E and R  s) and (E) = P(T  E)P(R  s). It is not hard to
see that they are both measures. Since they agree for sets E of the form {T  t} they must
agree for all Borel sets. Therefore P(T  E and R  s) = P(T  E)P(R  s) for all Borel
sets E. A similar argument shows P(T  E and R  F) = P(T  E)P(R  F) for all Borel
sets E and F. 
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Definition 2. If T and R are random variables, then the measure PT,R on RR induced by
T and R is
PT,R(E) = P{(T,R)  E}
= probability of the ordered pair of values (T, R) lying in E
Proposition 2. T and R are independent  PT,R = PT  PR where PT  PR is the
product measure of PT and PR.
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