2.9 Exercises 43(j)Is it true or not thatA,B,C are mutually independent if and only if
P(A∩B∩C) =P(A)P(B)P(C)? Prove your claim.2.11(Independence and random elements) Solve the following problems:(a)LetXbe a constant random element (that is,X(ω) =xfor anyω∈Ω over
the outcome space over whichXis defined). Show thatXis independent of
any other random variable.
(b)Show that the above continues to hold ifXis almost surely constant (that is,
P(X=x) = 1 for an appropriate valuex).
(c)Show that two events are independent if and only if their indicator random
variables are independent (that is,A,B are independent if and only if
X(ω) =I{ω∈A}andY(ω) =I{ω∈B}are independent of each other).
(d)Generalize the result of the previous item to pairwise and mutual independence
for collections of events and their indicator random variables.
2.12 Our goal in this exercise is to show thatXis integrable if and only if|X|
is integrable. This is broken down into multiple steps. The first issue is to deal
with the measurability of|X|. While a direct calculation can also show this, it
may be worthwhile to follow a more general path:
(a) Anyf:R→Rcontinuous function is Borel-measurable.
(b) Conclude that for any random variableX,|X|is also a random variable.
(c)Prove that for any random variableX,Xis integrable if and only if|X|
is integrable. (The statement makes sense since|X|is a random variable
wheneverXis).
Hint For (b) recall Exercise 2.1. For (c) examine the relationship between|X|
and (X)+and (X)−.
2.13(Infinite-valued integrals) Can we consistently extend the definition of
integrals so that for nonnegative random variables, the integral is always defined
(it may be infinite)? Defend your view by either constructing an example (if you
are arguing against) or by proving that your definition is consistent with the
requirements we have for integrals.
2.14 Prove Proposition 2.6.Hint You may find it useful to use Lebesgue’s dominated/monotone convergence
theorems.
2.15 Prove that ifc∈Ris a constant, thenE[cX]=cE[X](as long asXis
integrable).2.16Prove Proposition 2.7. Hint: Follow the ‘inductive’ definition of Lebesgue
integrals, starting with simple functions, then nonnegative functions and finally
arbitrary independent random variables.