Math 563 - Take Home Final Solutions
1. Let X and Y be independent random variables. Suppose that X is
discrete, i.e., it takes on at most countably many values, and Y has a density,
i.e., there is a non-negative measurable function f (x) on the real line
Math 563 - Take Home Midterm Solutions
1. Let f (x) be a Borel measurable real-valued function on [0, 1] with
12
f (x) dx < . Let cfw_Un be an independent sequence of random varin=1
0
ables, each of which is uniformly distributed on [0, 1], i.e., the dis
Math 563 - Homework 7
1. The gamma distribution is a two parameter family of distributions for
non-negative random variables. The parameters and are both positive
and the density is
f (x) =
1 x
xe,
( )
x0
where is the usual gamma function. (z ) = 0 xz 1
Math 563 - Homework 6 Solutions
1. (from Durrett) Let Xn be a sequence of integer valued random variables,
X another integer valued random variable. Prove that Xn converge to X in
distribution if and only if
lim P (Xn = m) = P (X = m)
n
for all integers m
Math 563 - Homework 5
1. Suppose Xn are real valued random variables dened on the same probability space. Prove that if Xn converges to 0 in probability, then there is a
subsequence that converges to 0 a.s. Hint: show there is a subsequence nk
such that P
Math 563 - Homework 4
1. Let X1 , X2 , , Xn be real-valued random variables on a common probability space. Prove they are independent if and only if
n
P (X1 x1 , X2 x2 , , Xn xn ) =
P (Xi xi )
i=1
for all x1 , x2 , , xn (, ].
SOLUTION: Recall that the den
Math 563 - Homework 3 - Selected solutions
2. (text, problem 14 on p. 64) Let X be a real valued random variable.
Let be an increasing real valued function on the real line. (Assume it is
Borel measurable.) Suppose that X and (X ) both have nite mean and
Math 563 - Homework 2 - Solutions
2. A sequence of real-valued RVs Xn is said to converge to the real-valued
RV X in probability if for every > 0 we have
lim P (|Xn X | > ) = 0
n
Show that if Xn converges to X almost surely, then Xn converges to X in
prob
Math 563 - Homework 1
1. Dene G = cfw_B F : X 1 (B ) F . Routine set manipulations show
that G is a -eld. By hypothesis, G contains E . So it contains the -eld
generated by E , which by hypothesis is F . This shows X is measurable.
2. First we note that t
Math 563 - Take Home Midterm
Due: Fri, Oct 19 4:00 pm in my oce or mailbox
The ne print: You are supposed to do this exam on your own. This means
you should not talk to anyone about the exam. You can ask me questions,
but I will only answer questions abou