Measure, Integration & Real Analysis, 2021a

Chapter 12 Probability Measures 399
11 Give an example of a probability space (Ω, F, P) and X, Y ∈L 2 (P) such
that σ 2 (X + Y) =σ 2 (X)+σ 2 (Y) but X and Y are not independent random
variables.
12 Prove that if X and Y are random variables (possibly on two different probability
spaces) and ˜X = Ỹ, then P X = P Y .
13 Suppose H : R → (0, 1) is a continuous one-to-one function satisfying conditions
(a) through (d) of 12.29. Show that the function X : (0, 1) → R produced
in the proof of 12.29 is the inverse function of H.
14 Suppose (Ω, F, P) is a probability space and X is a random variable. Prove
that the following are equivalent:
• ˜X is a continuous function on R.
• ˜X is a uniformly continuous function on R.
• P(X = t) =0 for every t ∈ R.
• ( ˜X ◦ X)˜(s) =s for all s ∈ R.
{
0 if x < 0,
15 Suppose α > 0 and h(x) =
α 2 xe −αx if x ≥ 0.
Let P = h dλ and let X be the random variable defined by X(x) =x for x ∈ R.
(a) Verify that ∫ ∞
−∞
h dλ = 1.
(b) Find a formula for the distribution function ˜X.
(c) Find a formula (in terms of α) for EX.
(d) Find a formula (in terms of α) for σ(X).
16 Suppose B is the σ-algebra of Borel subsets of [0, 1) and P is Lebesgue measure
on ( [0, 1], B ) . Let {e k } k∈Z + be the family of functions defined by the fourth
bullet point of Example 8.51 (notice that k = 0 is excluded). Show that the
family {e k } k∈Z + is an i.i.d.
17 Suppose B is the σ-algebra of Borel subsets of (−π, π] and P is Lebesgue
measure on ( (−π, π], B ) divided by 2π. Let {e k } k∈Z\{0} be the family of
trigonometric functions defined by the third bullet point of Example 8.51 (notice
that k = 0 is excluded).
(a) Show that {e k } k∈Z\{0} is not an independent family of random variables.
(b) Show that {e k } k∈Z\{0} is an identically distributed family.
Measure, Integration & Real Analysis, by Sheldon Axler