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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2008
Midterm Exam #2
November 13th, 6:008:00pm, Agricultural Engineering 119
Overview
The exam consists of ﬁve problems for 120 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 pageside 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. and have joint probability density function
¡ 1. The random variables
A(
"
#!©§ ¦¤¢
¨ ¥ £
4 0 8 8 0 4 0 0 ( $ $
BA@ 9765321)¤'&% otherwise [8] (a) Find the value of .
" . Y¨
8
C X
VU (
WAT
¡ 4U
cbA( e
gf¨ as:
(
D
A(
$
6' else d
r¨ q
s¦a¢ , the probability density function of .
p !ih¨
d . Find 8X
a%`E 4 004
BS2RQE ¡d
¨
p Let such that D 8 G
I¨ § H£ ¢ [10] (e) Deﬁne the function . Given this information, ﬁnd the C [7] (d) Somebody tells you that
is maximized. given
. For your
in terms of .
¡ ¨ D
)¡ [10] (c) Find
, the conditional probability density function of
limits (which you should not forget), use
and then bound
P F4
E [5] (b) Write an expression (no need to evaluate) for 2. Jointly Gaussian random variables:
, u
vt (
# w ¡ vt
u
u
vt
9 ¨ C ¡
E T
¡
¥
§ ¦£ y
w $ ¡ t
u [10] (b) Let and be jointly Gaussian random variables. Suppose you know that
, and that and are independent. By doing measurements, you ﬁnd that
and that
. Find possible values for the pair ( ,
), the variances of
respectively, and (do not forget this part) the correlation coefﬁcient
.
§ ¦£
¥ $§ $£ ¡ V
w $ ¡ (
B w
©$ ¡ ¡ t
u u
vt
, and
,
and , (where the Borel ﬁeld is D~ D
45l
0
i
D
~
40
5l
i
}0
D
~
0
!l
i0
}( d
¨ C $
$
u zw u t
y
$
u zw u t
y$
{zxvt
uy wu w eA(
4 oq r l i ¨
pnmkj¨
w
u C qs u
t . Find u $ Ti¨
Let g
hfd w eA(
4 3. [10] Let the probability space
be given by
restricted to
, of course) and, for any interval, , ¡ ¡
u
vt
yT w $ $ u
vt
w
(
` w
(
x ` . , be jointly Gaussian random variables. Let
. Deﬁne
. Find y
# w $ [10] (a) Let and
, and . 4. Counterexamples:
¨
¤& w 6mA(
[10] (a) Let the random variable
be uniformly distributed on
. Deﬁne
and
. Show that and are uncorrelated but not independent. Be rigorous here, particularly
for the “not independent” part.
u ¡ converges in mean square to
but does not converge with
, and the deﬁnition of the random variables
.)
¨ C v¨
d
k ¨
@Y ¡ [10] (b) Give an example where
probability one. (Be sure to give 5. Convergence:
w eA(
4 d w eA(
4 g
d w eA(
4 g ¨ 
)
d
¨ [10] (a) Let the probability space
be given by
,
(restricted to
, of
course), and
. For
, let
. Does this sequence of random
variables converge in some sense? If so, to what and in what senses (almost surely, in probability,
in mean square, in distribution)? As always, be sure to justify your claims. If you claim it does not
converge in any sense, just establish that it does not converge to a limit in one sense (almost surely, in
probability, in mean square, in distribution)  your choice!
(restricted to
u , w eA(
4 £Q4
¢ uE
P
u , of
$
D ¤
¥
$ C be given by
, let u
4 C ¡ i El
}h &mkj¨
l i¨
d
¨ C [10] (b) Let the probability space
course), and
. For ¡ i El l i¨
nmzj¨ C !I¨ Does this sequence of random variables converge in some sense? If so, to what and in what senses
(almost surely, in probability, in mean square, in distribution)? As always, be sure to justify your
claims. If you claim it does not converge in any sense, just establish that it does not converge to a
limit in one sense (almost surely, in probability, in mean square, in distribution)  your choice!
(where the Borel ﬁeld is d w eA(
4 , g
# u
B D$ D
45l
0
i
$
D
45l
0
i
}0
$
D
0
!l
i0
}(
c4 E ¨
¨ d
¨ [10] (c) Let the probability space
be given by
restricted to
, of course) and, for any interval,
C $
$
uxyzwxvt
u$
uy wu
{z$ xvt
u zw u t
y w eA(
4 oq r
qs l i¨
nmkj¨ u C A sequence of random variables is deﬁned as follows. If is a rational number,
for
all . If is an irrational number,
for all . Does this sequence of random variables
converge in some sense? If so, to what and in what senses (almost surely, in probability, in mean
square, in distribution)? As always, be sure to justify your claims. If you claim it does not converge in
any sense, just establish that it does not converge to a limit in one sense (almost surely, in probability,
in mean square, in distribution)  your choice! ¦ u ¢
R¨ ¦ ...
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 Fall '10
 DennisGoeckel

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