ENEE620
Homework 7 Solution
Problem 1. The random vector X = (X1 , X2 , X3 ) has
4 2
CX = 2
5
2
3
Due 11/25/2014
covariance matrix
2
3
7
(i) Determine a lower-triangular matrix A and its inverse A1 , such that the vector Z given by
X = AZ
Z = A1 X
has u

ENEE620
Homework 8 Solution
Due 12/04/2014
Problem 1. Let (X, Y ) be a zero-mean Gaussian vector with correlation matrix
1
1
,
where | < 1.
(i) Find the linear MMSE estimator of X 2 given Y and the corresponding MSE.
Solution: Note that E[X 2 Y ] = 0 due

ENEE620
Homework 5 Solution
Due 11/04/2014
Problem 1. The following parts are independent:
(i) Let X, Y L2 (, F, P). If E[X|Y ] = Y and E[Y |X] = X, then show that P(X = Y ) = 1.
Solution: We have:
E[XY ] = E[E[XY |Y ] = E[Y E[X|Y ] = E[Y 2 ]
Similarly,
E

ENEE620
Homework 5
Due 11/04/2014
Problem 1. The following parts are independent:
(i) Let X, Y L2 (, F, P). If E[X|Y ] = Y and E[Y |X] = X, then show that P(X = Y ) = 1.
(ii) Let X be exponential with parameter = 1. For t > 0, let Y1 = mincfw_X, t and Y2

ENEE 620: Random Processes in Communication and Control
Electrical and Computer Engineering Department
University of Maryland College Park
Fall 2011
Professor John S. Baras
SOLUTIONS to MIDTERM EXAM 1
November 2, 2011
Problem 1 (easy) (10 points)
A robot

ENEE 620
RECITATION 3
1. Let U and S be two discrete sets, and let f be a mapping S U S. Let (Un , n 1) be an
i.i.d. sequence with range U, and let X0 be independent of the sequence (Un ).
(i) Show that the sequence (Xn , n 1) dened iteratively by
Xn = f

ENEE 620
RECITATION 7
1. Let X be uniformly distributed over [1/2, 1/2). Derive and sketch its moment generating
function MX (s) and characteristic function X (u). How would you approximate both functions by
quadratic polynomials in the vicinity of the or

ENEE 620
RECITATION 6
1. Let X be a nonnegative random variable whose distribution is absolutely continuous except
possiblyfor a discrete mass at the origin.
(i) Show that
E[X] =
0
(1 FX (x) dx
(ii) How are the cdfs of X and Y = X related? Derive an expre

ENEE 620
RECITATION 4
1. Consider the Markov chain X0 with state space S = cfw_1, 2, 3, 4, 5 and transition probability
matrix
1
0
0
0
0
1
0
0
0
1
0
0
P =
,
0
0
1
0
0
0
1
where all entries containing letters are nonzero. If the distribution of X0 is uni

ENEE 620
RECITATION 5
1. (i) Suppose that X has absolutely continuous distribution, and let
Y = mincfw_|X|, 1
Determine the cdf and pdf of Y in terms of the cdf and pdf of X.
(ii) A fair coin is tossed. If it comes up heads, we set W = X (as above). Other

Convergence of Random Variables Review
Let = [0, 1) with (as usual) F being the Borel -eld and P [ ] the Lebesgue measure. For each
of the following sequences X1 , determine which modes of convergence are applicable and the limit
variable X (or distributi

ENEE 620
RECITATION 2
1. A coin whose sides are labeled 0 and 1 is tossed an innite number of times. The probability of
1 (in any particular toss) is given by p (0, 1). Denote the resulting innite sequence of 0s and
1s by .
(i) Let M be a xed integer, and

ENEE 620
RECITATION 1
1. A biased coin has P [H] = p, where 0 < p < 1. It is tossed an innite number of times. Let A
be the event that, starting at some point, the sequence of outcomes exhibits periodic behavior, i.e.,
a certain string (of arbitrary lengt

Appendix of recitation 2
Consider the i.i.d. sequence X1 , X2 , X3 , . . . of random variables such that Xi cfw_1, 2, 3, . . . and
(i)
P (Xn = i) = pi > 0
Let Yn = 1 with probability 1. For n 2, let Yn = 1 if the value of Xn has not been observed
previous

ENEE 620
RECITATION 8
1. Let X1 be i.i.d. uniform over (0, 1].
(i) What is the almost sure limit of
X1 + + Xn
Xn =
?
n
(ii) If Xi = Xi Icfw_Xi 2/3 , what is the almost sure limit of
Vn =
X1 + + Xn
?
n
(iii) What is the almost sure limit of Yn = (X1 X2 Xn

Transience and Recurrence (Persistence) of States
For a time-homogeneous Markov chain X0 , X1 , X2 , with a countable state space S, a state i is
said to be recurrent or persistent, if with probability 1 there exists some nite n, such that Xn = i
given th

Important Corollary to the Markov Property
Corollary to the Markov Property: For a Markov chain X0 , X1 , X2 , with a countable
state space S, we have:
n+m
n1
n+m
P Xn+1 A|Xn = sn , X0 B = P Xn+1 A|Xn = sn
for all A S m , B S n , and sn S.
The following s

Linear Prediction of WSS Random Processes
In order to construct the mathematical framework for prediction of WSS processes, we went through
the following steps:
Step 1. Choosing L2 as a function space endowed with inner product. For X, Y
L2 (, F, P ), we

Cherno Bounds and Bernsteins Inequality
Consider a sequence of random variables X1 , X2 , which are zero-mean and i.i.d. with a moment
generating function MX (s) dened over s (s0 , s0 ). We want to establish the following bounds,
known as the Cherno bound

Proofs of The Borel-Cantelli Lemmas
The First Borel-Cantelli Lemma: For a countable sequence of events A1 , A2 , dened
over a probability space (, F, P ), we have:
P (An ) < = P
lim sup An
=0
n
n1
Proof. Recall the denition of limit superior:
lim sup An :

ENEE 620
RECITATION 9
1. The random vector X = [X1 X2 X3 ]T has mean zero and covariance
CX
2 1
= 1 2 1
1 2
(i) Express CX in the form CX = LLT , where L is a real-valued lower triangular matrix. What is
the range of allowable values of ?
(ii) What lin

ENEE 620
RECITATION 11
1. For what values of a is rn = (a + |n|)2|n| the autocorrelation function of some wide-sense
stationary seequence?
(Use
k
k=0 z
= (1 z)1 and
k
k=1 kz
= z(1 z)2 for |z| < 1.)
2. Consider the wide-sense stationary sequence X , where

ENEE 620
RECITATION 10
1. Consider a Gaussian vector X = (X1 , X2 ) with characteristic function
X (u) = exp j(u1 4u2 ) 4u2 6u1 u2 9u2
1
2
(i) Write an expression for the joint pdf fX1 X2 (x1 , x2 ).
(ii) Determine the characteristic function of Y = X1 X2

Convergence of Random Variables Review Quiz
Solutions
Let = [0, 1) with (as usual) F being the Borel -eld and P [ ] the Lebesgue measure. For each
of the following sequences X1 , determine which modes of convergence are applicable and the limit
variable X

Markov Chain Convergence Notes
X0 is an irreducible time-homogeneous Markov chain
S = Z or subset thereof
j is an arbitrary xed state in S
Xn ?
Positive Recurrent
(e.g., S nite)
Null Recurrent
(e.g., symmetric random walk)
Transient
(e.g., asymmetric r

Markov Convergence Review
X0 is an irreducible time-homogeneous Markov chain
S = Z or subset thereof
j is an arbitrary xed state in S
Xn ?
Positive Recurrent
Null Recurrent
Transient
Yn = Icfw_Xn =j ?
Zn =
n
k=1
Yk ?

ENEE620
Homework 6
Due 11/11/2014
Problem 1. X1 , X2 , . . . are i.i.d. exponential with unknown parameter . Let Xi be the
+Xn
integer part of Xi ; and let Yi equal 1 if Xi is odd, 0 if Xi is even. Let Xn = X1 +X2n
and Yn = Y1 +Y2 +Yn . Find real-valued f