Measure, Integration & Real Analysis, 2021a

386 Chapter 12 Probability Measures
If F is clear from the context, the phrase “random variable on Ω” can be used
instead of the more precise phrase “random variable on (Ω, F)”. If both Ω and F
are clear from the context, then the phrase “random variable” has no ambiguity and
is often used.
Because P(Ω) =1, the expectation EX of a random variable X ∈L 1 (P) can be
thought of as the average or mean value of X.
The next definition illustrates a convention often used in probability theory: the
variable is often omitted when describing a set. Thus, for example, {X ∈ U} means
{ω ∈ Ω : X(ω) ∈ U}, where U is a subset of R. Also, probabilists often also omit
the set brackets, as we do for the first time in the second bullet point below, when
appropriate parentheses are nearby.
12.14 Definition independent random variables
Suppose (Ω, F, P) is a probability space.
• Two random variables X and Y are called independent if {X ∈ U} and
{Y ∈ V} are independent events for all Borel sets U, V in R.
• More generally, a family of random variables {X k } k∈Γ is called independent
if {X k ∈ U k } k∈Γ is independent for all families of Borel sets {U k } k∈Γ in R.
12.15 Example independent random variables
• Suppose (Ω, F, P) is a probability space and A, B ∈F. Then 1 A and 1 B are
independent random variables if and only if A and B are independent events, as
you should verify.
• Suppose Ω = {H, T} 4 is the sample space of four coin tosses, with Ω and P as
in Example 12.8. Define random variables X and Y by
X(ω 1 , ω 2 , ω 3 , ω 4 )=number of ω 1 , ω 2 , ω 3 that equal H
and
Y(ω 1 , ω 2 , ω 3 , ω 4 )=number of ω 3 , ω 4 that equal H.
Then X and Y are not independent random variables because P(X = 3) = 1 8
and P(Y = 0) = 1 4 but P({X = 3}∩{Y = 0}) =P(∅) =0 ̸= 1 8 · 14 .
• Suppose (Ω 1 , F 1 , P 1 ) and (Ω 2 , F 2 , P 2 ) are probability spaces, Z 1 is a random
variable on Ω 1 , and Z 2 is a random variable on Ω 2 . Define random variables X
and Y on Ω 1 × Ω 2 by
X(ω 1 , ω 2 )=Z 1 (ω 1 ) and Y(ω 1 , ω 2 )=Z 2 (ω 2 ).
Then X and Y are independent random variables on Ω 1 × Ω 2 (with respect to the
probability measure P 1 × P 2 ), as you should verify.
If X is a random variable and f : R → R is Borel measurable, then f ◦ X is a
random variable (by 2.44). For example, if X is a random variable, then X 2 and e X
are random variables. The next result states that compositions preserve independence.
Measure, Integration & Real Analysis, by Sheldon Axler