Measure, Integration & Real Analysis, 2021a

Chapter 12 Probability Measures 387
12.16 functions of independent random variables are independent
Suppose (Ω, F, P) is a probability space, X and Y are independent random
variables, and f , g : R → R are Borel measurable. Then f ◦ X and g ◦ Y are
independent random variables.
Proof Suppose U, V are Borel subsets of R. Then
P ( { f ◦ X ∈ U}∩{g ◦ Y ∈ V} ) = P ( {X ∈ f −1 (U)}∩{Y ∈ g −1 (V)} )
= P ( X ∈ f −1 (U) ) · P ( Y ∈ g −1 (V) )
= P( f ◦ X ∈ U) · P(g ◦ Y ∈ V),
where the second equality holds because X and Y are independent random variables.
The equation above shows that f ◦ X and g ◦ Y are independent random variables.
If X, Y ∈L 1 (P), then clearly E(X + Y) =E(X)+E(Y). The next result gives
a nice formula for the expectation of XY when X and Y are independent. This
formula has sometimes been called the dream equation of calculus students.
12.17 expectation of product of independent random variables
Suppose (Ω, F, P) is a probability space and X and Y are independent random
variables in L 2 (P). Then
E(XY) =EX · EY.
Proof First consider the case where X and Y are each simple functions, taking
on only finitely many values. Thus there are distinct numbers a 1 ,...,a M ∈ R and
distinct numbers b 1 ,...,b N ∈ R such that
X = a 1 1 {X=a1 } + ···+ a M 1 {X=aM } and Y = b 1 1 {Y=b1 } + ···+ b N 1 {Y=bN } .
Now
Thus
XY =
M N
∑ ∑
j=1 k=1
E(XY) =
a j b k 1 {X=aj } 1 {Y=b k } =
=
M N
∑ ∑
j=1 k=1
M N
∑ ∑
j=1 k=1
a j b k 1 {X=aj }∩{Y=b k } .
a j b k P ( {X = a j }∩{Y = b k } )
( M )( N )
∑ a j P(X = a j ) ∑ b k P(Y = b k )
j=1
k=1
= EX · EY,
where the second equality above comes from the independence of X and Y. The last
equation gives the desired conclusion in the case where X and Y are simple functions.
Measure, Integration & Real Analysis, by Sheldon Axler