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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2006
Midterm Exam #2
November 13th, 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 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. 1. You are on your ﬁrst job and are asked to characterize a random variable . By making a large
number of observations, you form a normalized histogram (it integrates to 1) as shown below. Using a
computer program, you ﬁt the following function to your histogram (i.e. this function is your estimate
of the probability density function of ): &
% $
¨ ¦¤ ¢
#!
" ©§¥£¡
13
54 1
2 )
0( ¨ ¦¤ ¢
©§¥'¡ )
0( (i.e. be deﬁned by:
FI
QP FD
GE 7 , the cumulative density function (CDF) of .
7 B
HF
@8
A9
B
C 7 ¨U
V§¤ S
TR Find ) [10] (b) Let the random variable , and Var 1
6 [10] (a) Use the estimated probability density function
to ﬁnd
,
the expected value of , the mean squared value of , and the variance of ). 0.25 0.2 f(x) 0.15 0.1 0.05 0
x 2 4 Figure 1: Normalized histogram of ) 1W
YXB ¦ . Given , the marginal probability density function of
b
f7 , the probability that 8 10 for Problem 1. c
egW c
¨ edW ¨U
V§¤
b
97 S ¡
¤ ) [12] (a) Find
a [8] (b) Find is uniform on 1WB
YXHF 2. Suppose that the random variable
variable on
. 6 . 7 −2 −4 . ¦
` −6 ,
7 −8 0
−10 is a uniform random 3. A coin is ﬂipped three times. Let the random variable be the number of heads in the ﬁrst two ﬂips
of the coin, and let the random variable be the number of tails in all three coin ﬂips.
7 and . The easiest way to express your answer is probably a ¨
VU
7 , the covariance of and .
7 ¦ B¦
¨
7 U
a ¤ B
C ¤ ¦ [7] (b) Find cov for all U [8] (a) Find
table in and . 4. The money (in thousands of dollars) made from investing in stocks “Ystock” and “Zstock” are modeled as the random variables and , respectively. Assume and are independent with respective
probability density functions
and
as shown below:
¡ ¡ 7 ¨ ¤¤ ¢
'¥£¡ ¨ ¥£¡
¤¤ ¢ ¨U
V§¤ S 7 ¡ ¨U
V§¤ §
© S ¡ §
©
@ @ ¨ ¨
W ¤¦
W
3 W U¦
W
3
[5] (a) You want to make as much money as possible, of course. Which stock would you buy?
[10] (b) Suppose you decide to buy “Ystock” and your friend decides to buy “Zstock”. What is the
probability that you make more money? (In other words, ﬁnd
).
¨¡ b 7 ¤ a 5. Convergence of random sequences:
¤
¥ 1WB
YXHF cW
¤
1WB
YXHF ¥¨
1 ¨ HF
B ¢ ¨ ¢
£B ¡¤ B [10] (a) Let the probability space
be given by
,
(restricted to
, of
course), and
. For
, let
. Does this sequence of random
variables converge? If so, to what and in what sense (almost surely, in probability, in mean square, in
distribution)? If not, just establish that it does not converge to a limit in one sense (almost surely, in
probability, in mean square, in distribution)  your choice!
) ¨ ¤
1 ¨ HF
B ¢ ) ¨ ¤
a ¨ c ¦
¨ ¨¦
g¨ ¨ ©B §¤ ¤ a ¢
£B ¡¤ B [10] (b) Let the probability space
be given by
,
(restricted to
, of
course), and
. For
, let
. Does this sequence of random
variables converge? If so, to what and in what sense (almost surely, in probability, in mean square, in
distribution)? If not, 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
) ¤
¥
¢ , 1WB
YXHF 1WB
YXHF
) ¤
, of
I % ¨ H¦
¨
¨
a
( I
¨ c ¦ ¨ ¨¦
¤
A ¨ ¨ ©B §¤ ¤ a ""
""
""
""
""
"" ""
"" ""
"" #
"" ¨
¤
"" ( B
a "" B 3 (
% ¨¦
¤
` ¨ ¨ ©B §¤ ¤ "" &&
''& ¨ "
!"" B 3
% c ¢
£B ¡¤ B B
HF a F
% I I 3 be given by
, let ) [10] (c) Let the probability space
course), and
. For ""
""
""
""
""
"" &&
''& W
% %
B
""
"$ Does this sequence of random variables converge? If so, to what and in what sense (almost surely, in
probability, in mean square, in distribution)? If not, just establish that it does not converge to a limit
in one sense (almost surely, in probability, in mean square, in distribution)  your choice! 6. Your team wins each match with probability 0.50 and loses each match with probability 0.50 (there
are no ties). Assume that each match is independent of all other matches. Answer the following
questions using the Central Limit Theorem.
[10] (a) Assuming that the season lasts 100 matches, ﬁnd the probability that your team wins more
than 55 matches.
[10] (b) Suppose that you wanted it such that your team wins at least 65 matches with probability
0.90. What is the minimum number of matches in a season for this to be true? Hint: Feel free to use
the Central Limit Theorem loosely as on the homework to solve this part. ...
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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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