Page 49 - Elements of Distribution Theory

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1.9 Exercises 35

1.2 Let A 1 \ A 2 denote the elements of A 1 that are not in A 2 .
(a) Suppose A 2 ⊂ A 1 . Show that
P(A 1 \ A 2 ) = P(A 1 ) − P(A 2 ).
(b) Suppose that A 2 is not necessarily a subset of A 1 . Does
P(A 1 \ A 2 ) = P(A 1 ) − P(A 2 )
still hold?
1.3 Let A 1 A 2 denote the symmetric difference of A 1 and A 2 ,given by
A 1 A 2 = (A 1 \ A 2 ) ∪ (A 2 \ A 1 ).
Give an expression for P(A 1 A 2 )in terms of P(A 1 ), P(A 2 ), and P(A 1 ∩ A 2 ).
1.4 Show that
c
P(A 1 ) ≤ P(A 1 ∩ A 2 ) + P(A ).
2
1.5 Show that
(a) P(A 1 ∪ A 2 ) ≤ P(A 1 ) + P(A 2 )
(b) P(A 1 ∩ A 2 ) ≥ P(A 1 ) + P(A 2 ) − 1.
1.6 Find an expression for Pr(A 1 ∪ A 2 ∪ A 3 )in terms of the probabilities of A 1 , A 2 , and A 3 and
intersections of these sets.
1.7 Show that (P3) implies (P2).
1.8 Let and P denote the sample space and probability function, respectively, of an experiment
and let A 1 , A 2 ,... denote events. Show that

∞
∞

Pr A n ≤ P(A n ).
n=1 n=1
1.9 Consider an experiment with sample space = [0, 1]. Let D(·) denote a function defined as
follows: for a given subset of , A,
D(A) = sup |s − t|.
s,t∈A
Is D a probability function on ?
1.10 Let X denote a real-valued random variable with distribution function F. Call x a support point
of the distribution if

F(x + ) − F(x − ) > 0 for all > 0.
Let X 0 denote the set of all support points of the distribution. Show that X 0 is identical to the
minimal support of the distribution, as defined in this chapter.
1.11 Prove Corollary 1.1.
1.12 Let X 1 and X 2 denote real-valued random variables with distribution functions F 1 and F 2 ,
respectively. Show that, if

F 1 (b) − F 1 (a) = F 2 (b) − F 2 (a)
for all −∞ < a < b < ∞, then X 1 and X 2 have the same distribution.
1.13 Let X denote a real-valued random variable with distribution function F such that F(x) = 0 for
x ≤ 0. Let
1/x if x > 0

f (x) =
0 otherwise
and let Y = f (X). Find the distribution function of Y in terms of F.