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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2002
Midterm Exam #1
October 17th, 6:008:00pm, Marston 132
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 one 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. A number is chosen at random from the interval [0,1]. As is the standard case, the probabilities are deﬁned on the Borel algebra (restricted to
[0,1]). Starting from ﬁrst principles (i.e. deﬁnition of the Borel algebra,
axioms of probability, etc.), answer the following three parts: ¡ [10] (a) Let be a subset of
that is not in the Borel algebra. Show
that must contain an uncountable number of elements. ©§¥£
¨¦¤¢ . Is , the comple ©§¥£
¨¢ ¡ [5] (b) Let be an arbitrary uncountable subset of
ment of , necessarily countable?
,
@9 ; that is,
©§¥£
¤¢ [10] (c) Let
be the set of irrational numbers in
if and only if
and
for all
. Find the probability of .
76 6 6 ¥ 5 ¥ 3 ¥ § ¥ £ 0
8¨¨4¦42¨1 (' ) $%
& §
#"!
£ 76 6 6 ¥ 5 ¥ 3 ¥ § ¥ £
8¨¨4BA2¨10 2. [10] A number is chosen from the interval [0,1] such that the likelihood of a
given result is proportional to the value . Deﬁne a nontrivial probability
, where is the observation
space for this experiment; that is, ﬁnd
space, is a set of subsets of to which probabilities are assigned, and
is a probability mapping from to
.
D Q
R4IHFEC
P¥
G¥
D P D ©§¥£
¨¢ G G 3. Let the continuous random variable be uniformly distributed between
and . Let the discrete random variable have probability mass function
deﬁned by
,
, and
. Form the
random variable
. Assume and are independent.
£ S T & TC
cbP T T
gS Q
V§ & TC
EUP f h
if Q tC
us1q
r
©
w¤Uv
f¢ T
f TC
EUP S & and Var & Q
23 Y
W
X& , the probability density function of 3 § & , what is the probability that . Q
d5 a
W
`& [10] (b) Find ©
R¤¢
f W
p [10] (c) Find e
W
X& [10] (a) Given that . . have joint probability density function: § h S h
"f h 6 £ and
f §
¨ ¥Q
AV§ S 4. The random variables else ¥
¦¤
C & Qt
H¥ C r ¥
£ ¢ £¡q ¥ [5] (a) Compute the constant . t
uC r
q © &
S f
¤¢ and Var © &
S f
¢
v [10] (d) Compute
at
. .
Q ©q F
C [5] (c) Compute the conditional probability density function Q [5] (b) Compute the marginal probability density function . . Interpret your answers £ ¢
Uv , the cumulative distribution function "!
#£ . Compute .
Q
dC W &
W
f S & [10] (f) Let
(CDF) of . . Find © & [5] (e) Let T T 666¥3
¨¨¦A¥ be mutually indepen 6
£ 8
9S &$
'¥ %S 6
7
£
h 0 420
531¤
¥ & , ..., . Find Q tC
us1q
r & § Q C Y ¥
¦£
S , (
)¥ W S ¡q and deﬁne as the maximum of
bility density function of . 7 5. [15] Let the random variables
dent and each have probability density function: , the proba f f ...
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 Fall '10
 DennisGoeckel

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