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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2002
Midterm Exam #2
November 20th, 6:008:00pm, Marston 132
Overview
The exam consists of six 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 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. 1. [10] A point is chosen at random within a circle of radius 3 (i.e. all points
within the circle are equally likely). The outcome (or observation) for the
experiment is the distance of the point from the center of the circle. Deﬁne
a nontrivial probability space for this experiment; that is, ﬁnd
,
where is the observation space, is a set of subsets of to which probabilities are assigned, and is a probability mapping from to
.
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¨ 2. A fair sixsided die is rolled once. Let be the number of spots appearing
on the top of the die. A coin is then ﬂipped
times, and the number of
heads (call it ) is recorded.
. , ﬁnd the probability that %
"$ !
# [7] (b) Given that !
" [8] (a) Find the probability that . 3. Suppose that your goal is to maximize the proﬁt of your business. If you
decide to travel to Xville, the proﬁt (in dollars) for your business is a random
as given below. If
variable with cumulative distribution function
you decide to travel to Yville, the proﬁt (in dollars) for the trip is a random
variable with cumulative distribution function
as given below. The
random variables and are independent.
0
21 (
)' 6
87 & 4
5' 3 3 6
B¢ & 0
21 4
A' (
9' 1 1 @
@
@ 2 3 0 1 1 2 3 [8] (a) To which city (Xville or Yville) do you travel to maximize the proﬁt
for your company. Be sure to justify your answer.
[7] (b) Suppose you have some extra time so you travel to both Xville and
Yville. Let the random variable be your total proﬁt from the two trips.
Sketch the cumulative distribution function
for the
random variable .
GE I C H¨ E
GF C D
5' E ¨£ ¥£ ¡
©§¦¤F T VUH¨ 4. [20] An experiment is deﬁned by the probability space
, is the Borel algebra restricted to
, and , where
is deﬁned 6 @ 1
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S ¥
R£
QP ¡ ¤
¥¢ ¡ ¢
££ ¡ H¨ by
. A sequence of random variables is deﬁned as follows.
for all . If is an irrational number,
If is a rational number,
for all . Does this sequence of random variables converge? If
so, in what ways and to what limiting random variable does it converge?
¦ © ¦ §
¨& © § ﬂip %$
& & ©
!
3
. .
©
) ©
! 3 3 . §
&
&
" ©
!
3 , the autocorrelation function of
, the mean function of "
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! £ © 4
§' '
) (
4 [10] (c) Find . If the
& §
[5] (a) Find the probability mass function of
[5] (b) Find ﬂip is “heads”, I let
.
5. I ﬂip a coin repeatedly. If the
is “tails”, I let
. Let [10] (d) Does the sequence of random variables
converge? If so, in
what ways and to what limiting random variable does it converge? [Be sure
to justify your answer].
©
) 3 0
1 0
1 and
be zeromean, widesense stationary Gaussian random
6. Let
processes that are independent of one another. (Recall that a Gaussian random process is one for which any collection of samples is jointly Gaussian). Assume that the two processes have the same autocorrelation function
. (Note that
). Deﬁne the
random process
as: 0 ( ( R S R `X P 0
Ucb aYH 1 0
3 C 0
1 V
W 0
3
C 0
21 . . F9 f
D 2
3 ( ( C 0
1 &
( 4 0
1 RS
UTR [email protected] 3 PIG 0
2
3 DC
EA
F D C A9 86 4
£EB@!75! . , the autocorrelation function of 0
1 d0
eY£ C 0
3 % '
D
D
¤( [10] (c) Find the probability density function & [8] (b) Find , the mean function of ! 3 [5] (a) Find d
e0 0 d
e0 0 [7] (d) Pick any two times and ,
, that you want (your choice!)
and give the joint probability density function
of the random variables
and
. (For example, you might choose
and
; then, you just have to give
).
5% 0 d
50 £ % 0
1 50 £
d
F 9 1F i 9
ph
Df
D g
% % % 0
7 D Fr pFq
es9 1eB9 D f d0
e3 C % 0
3 C !
# d
e0 ...
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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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