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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Midterm Exam #2
November 22nd, 6:008:00pm, Goessman 20
Overview
The exam consists of four problems for 130 points. The points for each part of each problem are given
in brackets  you should spend your two hours accordingly. The exam is closed book, but you are allowed two pagesides of notes. Calculators are not allowed.
I will provide all necessary blank paper. Testmanship
Full credit will be given only to fully justiﬁed answers. Giving the steps along the way to the answer will not only earn full credit but also maximize the
partial credit should you stumble or get stuck. If you get stuck, attempt to neatly deﬁne your approach
to the problem and why you are stuck. If part of a problem depends on a previous part that you are unable to solve, explain the method for
doing the current part, and, if possible, give the answer in terms of the quantities of the previous part
that you are unable to obtain. Start each problem on a new page. Not only will this facilitate grading but also make it easier for you
to jump back and forth between problems. If you get to the end of the problem and realize that your answer must be wrong, be sure to write “this
must be wrong because . . . ” so that I will know you recognized such a fact.
Academic dishonesty will be dealt with harshly  the minimum penalty will be an “F” for the course. You may ﬁnd the following fact useful as you solve this exam:
Fact: Given a collection of jointly Gaussian random variables, every linear combination is Gaussian.
Conversely, given that every linear combination of a collection of random variables is Gaussian, that collection of random variables is jointly Gaussian. have joint probability density function:
3 ) ) 63 ) )
@91%8754210( and
¡ 1. The random variables
A(
"
#!©§ ¦¤¢
¨ ¥ £
$
'&%¨ otherwise [10] (a) Find .
" ¨ £
DCB¢
G E
I H¨ £ F§ ¢ , the conditional probability density function of given . For this, write
between constants and your limits on as (potentially) functions of .
¡ [5] (c) Find
your limits for . , the marginal probability density function of [5] (b) Find [5] (d) Suppose that you are planning to estimate from an observation . Find the such that when
, you are “best” able to estimate from
. In other words, if you are trying to estimate
and are hoping for an
that makes it easy, what would you choose?
¡ ¡ (Note: Some of these answers are short):
, let
W , let be ¡ ¡ U
V¡ ¨ G
S G T [5] (b) Let
be a random variable uniformly distributed between 0 and 1. Given
Gaussian with mean and variance . Find
.
! be a Gaussian random variable with mean 0 and variance 1. Given
[7] (a) Let
uniformly distributed between 0 and . Find
. ¡ ¡
P
R and
Q 2. More multiple random variables be U
V¡ ¨ T X be a random variable uniformly distributed between 0 and 1. Let let be Gaussian
[8] (c) Let
with mean and variance . Suppose and are independent. Find
(you can leave an
integral expression here).
(restricted to h3
igA( e T ¡ q
rpc igA( fd`
h3 e . ¨ U
V¡ ¨
T T X s
v
a¨
8
y
8
c `
2ba¨ . Find U
Y¡ 8 ( " v s¨
xwgut¨ T
Rx¨
¡
Rx¨ " Suppose , Find . . be given by ¡ [10] (d) Let the probability space
course), and , of 3. Suppose that I am interested in characterizing two random variables and (both individually and
their relation to each other). I know in advance that the variances of the two random variables are
equal (
), but I do not know . I know nothing about either mean, including whether
they are equal or not. Unfortunately, I am not able to observe the random variables individually, but
only
and
. Making a bunch of measurements, I come up with the following:
8 8 y
Ih £
¤Xy .
.
$ ,
and
, respectively. Answer these © ¡ $ ¡ ¡¢
¡¢
¡¢
¡
¢ e e ¨e ¨e ¡ 8 $ £ [15] (c) You have some extra time on your hands, so you deﬁne
and
, the probability density functions and
estimate
three parts separately: and ¡ § x£
¥ b¨ ¢
§ , the correlation coefﬁcient of ¡ ¡ ¡ h e
¡
h
8¡y
h
$ ¥
¦X e
Ih and y ¡ ¡y and .
.
h 8 8 ¡ .
XX § [10] (b) Find 8 [5] (a) Find ¡ . ¨ © ¨ ©
x¢
¨ ¨ Suppose that you note that is Gaussian and is not Gaussian. Are and jointly Gaussian?
(possible answers are yes, no, or maybe  a short justiﬁcation is ﬁne.)
Suppose that you note that is Gaussian and
is Gaussian. Are and jointly Gaussian?
(possible answers are yes, no, or maybe  a short justiﬁcation is ﬁne.)
Suppose that you note that
and
are jointly Gaussian. Are
and
jointly Gaussian?
(possible answers are yes, no, or maybe  be rigorous here with your justiﬁcation.)
¡ ¡ ¡ , of course),
.
8 © $ #Fx¨
¨
X
h A( e
¨ © q X
c h A( e
4` ¨ c `
2ba¨ 4. (a) Let the probability space
be given by
,
(restricted to
and
. Let a sequence of random variables be deﬁned by
8
T s G 8 ¨
y v v s¨
x g t¨ T #!
$ E£ "£ ¢ [8] Find
, the conditional probability density function of
given
.
[12] Determine whether the sequence
converges, and, if so, to what and in what ways?
Consider almost sure convergence, mean square convergence, convergence in probability, and
convergence in distribution. You can do them in any order that you want, and you can use one
type of convergence to imply another when appropriate. x¨
X
X
h
i3 53
e
y &
' % y
53 e
@` be given by
,
(restricted to
. Let a sequence of random variables be deﬁned by ((
) ) q
RWc i3
h ((
)X )X , T
s v g t¨
v s¨
y
c `
2ba¨ 1(
53 8 2)X
3
4 ¨ 1(
2)X y (
R ¨
D¨ T #£
0B¢ [5] Find
, the probability density function of
[10] Determine whether the sequence
converges, and, if so, to what and in what ways?
Consider almost sure convergence, mean square convergence, convergence in probability, and
convergence in distribution. You can do them in any order that you want, and you can use one
type of convergence to imply another when appropriate. &
' % (b) Let the probability space
of course), and
. y X ¨ [10] (c) Let
be integer random variables such that
.
. Determine whether the sequence
converges, and, if so, to what and in what ways?
Consider mean square convergence, convergence in probability, and convergence in distribution. You
can do them in any order that you want, and you can use one type of convergence to imply another
when appropriate.
T X T &
9 % 1 887g X
666 3 ...
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This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).
 Fall '10
 DennisGoeckel

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