Homework Set 1
1. For two events A and B , show that Ac B c = AB .
2. If an > 0, show that [0, an ) = [0, supn an ) but [0, an ] may
not be the same as [0, supn an ].
3. Let An = [1/n, n] for n even, An = [n, 1/n] for n odd. Find
lim sup An and li
Homework Set 2
1. Let A1 , . . . , An be arbitrary subsets of a set . Find an attainable
upper bound for the cardinality of A1 , . . . , An .
Considering the case n = 2, show that your bound is actually attained.
2. Let C be a countable semield on a space
Homework Set 4
1. Let P be a probability measure on (, A ). Let E , , be disjoint
sets from A . Show that cfw_ : P (E ) > 0 is at most countable.
2. Let P be a probability measure on (, A ). Two sets A and B are
called almost disjoint if P (A B ) = 0.
Homework Set 6
1. Let cfw_Xn be uniformly integrable and Xn = n1 n Xi . Show that
cfw_Xn is also uniformly integrable.
2. A sequence of random variables Xn is called uniformly integrable from
above if supn E(Xn 1lcfw_Xn > C ) 0 as C .
Show that if X
Homework Set 7
1. Let Fi be a eld on i , Ai = Fi , i = 1, 2. Let F1 F2 be the
eld generated by the semield R = cfw_F1 F2 : F1 F1 , F2 F2 .
Show that F1 F2 = cfw_k=1 Ri : R1 , . . . , Rk R , k 1.
Show that F1 F2 = A1 A2 , the product -eld.
2. For a nonne
Homework Set 8
1. Let cfw_Xn be a sequence of random variables on a probability space
(, A , P ). Show that Xn s are mutually independent if and only if X1 , . . . , Xn1
and Xn are independent for every n.
2. Let (X, Y ) have a bivariate normal distribut
Homework Set 9
1. Let X , Xn , n 1, be random variables dened on a probability space
(, A , P ) and that Xn p X . Suppose that there exists N A , P (N ) = 0
such that for all N , Xn ( ) is a monotone sequence of real numbers.
Then show that Xn X a.s.
Homework Set 10
Below E(Y |X ) means E(Y | X ).
1. Let X, Y be square-integrable random variables on a probability space
(, A , P ). If E(Y |X ) = X and E(X |Y ) = Y , show that X = Y a.s. [P ].
[Hint: Compute E(Y X )2 ]
2. If X is a square-integrable ran