Measure, Integration & Real Analysis, 2021a

388 Chapter 12 Probability Measures
Now consider arbitrary independent random variables X and Y in L 2 (P). Let
f 1 , f 2 ,... be a sequence of Borel measurable simple functions from R to R that
approximate the identity function on R (the function t ↦→ t) in the sense that
lim n→∞ f n (t) =t for every t ∈ R and | f n (t)| ≤|t| for all t ∈ R and all n ∈ Z +
(see 2.89, taking f to be the identity function, for construction of this sequence). The
random variables f n ◦ X and f n ◦ Y are independent (by 12.17). Thus the result in
the first paragraph of this proof shows that
E ( ( f n ◦ X)( f n ◦ Y) ) = E( f n ◦ X) · E( f n ◦ Y)
for each n ∈ Z + . The limit as n → ∞ of the right side of the equation above equals
EX · EY [by the Dominated Convergence Theorem (3.31)]. The limit as n → ∞
of the left side of the equation above equals E(XY) [use Hölder’s inequality (7.9)].
Thus the equation above implies that E(XY) =EX · EY.
Variance and Standard Deviation
The variance and standard deviation of a random variable, defined below, measure
how much a random variable differs from its expectation.
12.18 Definition variance; standard deviation; σ(X)
Suppose (Ω, F, P) is a probability space and X ∈L 2 (P) is a random variable.
• The variance of X is defined to be E ( (X − EX) 2) .
• The standard deviation of X is denoted σ(X) and is defined by
√
σ(X) = E ( (X − EX) 2) .
In other words, the standard deviation of X is the square root of the variance
of X.
The notation σ 2 (X) means ( σ(X) ) 2 . Thus σ 2 (X) is the variance of X.
12.19 Example variance and standard deviation of an indicator function
Suppose (Ω, F, P) is a probability space and A ∈Fis an event. Then
σ 2 (1 A )=E ( (1 A − E1 A ) 2)
= E ( (1 A − P(A)) 2)
= E(1 A − 2P(A) · 1 A + P(A) 2 )
= P(A) − 2 ( P(A) ) 2 +
( P(A)
) 2
= P(A) · (1 − P(A) ) .
√
Thus σ(1 A )= P(A) · (1 − P(A) ) .
Measure, Integration & Real Analysis, by Sheldon Axler