Math 671 - Assignment 6 - Due October 16
1. (Dont hand in.) If p > 0 and X Lp then xp P (X > x) 0 as x
2. If X, Y are independent and X + Y Lp (p > 0) then X Lp . (Hint:
Show that for large enough > 0,
P (|X| > ) 2P (|X| > , |Y | < /2) 2P (|X + Y | > /2)
Math 671 - Assignment 4 - Due September 27
1. Prove that if X N (0, 1) then X 2 (1/2, 2)
(Note: We say
Y (, ),
(, > 0)
provided Y has pdf
f, (t) =
t1 et/ It>0
where (x) = 0 tx1 et dt is the gamma function. You should be able
to do this without actua
Math 671 - Assignment 3 - Due September 20
1. Let (, A) be a measurable space, and f : any function. Prove
that f 1 (A) = cfw_f 1 A : A A is a algebra on .
2. (Dont turn in) (a) If a topological space (X, T) is second countable, then
the Borel algebra BX
Math 671 - Assignment 5 - Due October 7
(Well do some work in class on Oct. 2nd that might help with problems
1. A point is chosen at random from in the unit square, and its distance
X from the nearest side of the square is measured. What is the CDF
Math 671 - Assignment 3a - (also) Due
1. (Barndor-Nielsen) Let cfw_An nN be a sequence of events, not necessarily
independent. Suppose (i) P (An ) 0, and (ii) n P (An An+1 ) < .
Prove that P (An i.o.) = 0.
Why is this a generalization of Bore
Math 671 - Assignment 2 - Due September 13
1. Let (, A, ) be a nitely-additive measure space. Consider the conditions:
(a) If Fn F then (Fn ) (F )
(b) If Fn F then (Fn ) (F )
(c) If Fn then (Fn ) 0
Prove that if is nite then these three conditio
Math 671 - Assignment 10 - Due Nov. 25
1. Text E16.2 (Turn this in)
2. Text E16.1 (Dont turn this in, but compare to E16.2)
3. (a) E16.4. (b) Do this again without a contour integral. (Hint: clever
integration by parts.) Turn in at least one of these, a o
Math 671 - Assignment 7 - Due Oct. 23
In all the exercises except the last one, Xn is a sequence of random variables
and Sn denotes the sum Sn = X1 + + Xn .
1. If supn |Xn | Lp and Xn X a.s. then X Lp and Xn X in Lp .
2. For any sequence of random variabl
Math 671 - Assignment 11 - Due Dec. 6
Turn in problems 1, 2, 5, and 6.
1. Prove that the characteristic function for any distribution is uniformly
2. Let X1 , X2 , . . . be iid random variables with the (a, b) distribution. Show
Math 671 - Assignment 9 - Due Nov. 15
(So far the only one Im sure I want you to hand in is the third problem. I
will add one of the others, I havent decided which one yet.)
Turn in the rst and third problems.
1. Text E9.2
2. Show that if C D are sub-alge
Math 671 - Assignment 12 (Not due, but look at
A couple of these (or something very like them) will be on the exam. Most
are just exercises in understanding the notation.
3. If Xn and Yn are submartingales, so is Xn Yn = max(Xn , Y
Math 671 - Assignment 1 - Due September 6
1. Let be an uncountable set. Show that
A := cfw_A : A countable orA co-countable
is a algebra.
2. Let C P(), (C) := cfw_A P() : C A, A a algebra. Prove
that (C) is the smallest algebra containing C. (Note that yo
Math 671 - Assignment 8 - Due Nov. 1 or
1. Text E4.6
2. Text E4.7
3. Let Xn be iid random variables, K > 0 a constant, and let Yn = Xn I[|Xn |K] .
Suppose that the following three series converge:
P (|Xn | > K);
n E(Yn );