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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2008
Midterm Exam #1
October 15th, 6:008:00pm, Agricultural Engineering 119
Overview
The exam consists of ﬁve 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 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. I have analyzed two independent experiments, Experiment 1 and Experiment 2, to arrive at two sepaand
, where:
rate probability spaces:
§¥ ¡
"!¥ ¨¢ @
BA9! is deﬁned by , , G
HFED9d
$ 5 is deﬁned by , 0R G
XSQP63I9!
$ 2 0 & 220 (&¥20&¥2(&¥ 5&
)31¥ )893896)876'%4§
$£
(
a`Y¢
$ ¡ £ 20 (&
31¥ )'%#¡
$£ WR G
XSV63I
$ 2 0 & £
2 T¥ 0&
S¨c'b¡
$ 2 2 T¥ 0&¥ 2 T&¥ 2 0&¥ 5&
)S¨c89S893876'b§
$
(
a`Y¢
$¡ , , and @ £
BA , , and G
HFEDC
$ 5 £ , TR G
USQP66)I
$ 2 ( & £ £ ¥£ §¥£ ¡
©¨¦¤¢ , , . . eR G
XSV6SI
$ 2T& [10] (a) Are these valid probability spaces? Be sure to tell me all of the conditions that you checked
to arrive at your answer.
[10] (b) My boss asks me to deﬁne a combined experiment as follows: Perform Experiment 1 and remember the result; then, perform Experiment 2 and remember the result. Now, write in your notebook
the outcome of the combined experiment as an ordered pair with the ﬁrst entry equal to the result of
Experiment 1 and and the second entry equal to the result of Experiment 2. For example, an outcome
for the combined experiment. Use a that captures as many events
might be “(1,4)”. Find
as possible, and be sure to write out explicitly at least half of the events in .
§ ¥ §¥ ¡
gf¨#¢ § [10] (c) Alas, the boss is ﬁckle and changes his mind. Now he asks: perform Experiment 1 and remember the result; then, perform Experiment 2 and remember the result. Now, write in your notebook
the outcome of the combined experiment as a random variable
equal to the result of Experiment
). Find
1 times the result of Experiment 2. For example, an outcome might be “4” (which is
for the random variable . Hint: Feel free to deﬁne using the integral of a function if
this makes it easier to represent.
h 0i
qp0 ¥ §¥ ¡
g4¨r¢ h Now, your buddy in the modeling department comes to you with yet another experiment description:
, where
s ¥s § s ¡
"!6t¨¥ ¢ @ s
BA9! , and is deﬁned by $ 2 ( & ¥ G
%66)I©SHPED©
$ 5 22 u¥ 0 (&¥2u¥ 0&¥2(&¥ 5&
yc1S1¥ )8931c896)8x6wv§
$s , . 2u¥ 0 (&
31c1¥ )'b¡
$s (
a631c1¥ )I©cXS631cI4¥ SG
$ 2 u ¥ 0 ( & ¥ R G
$ 2 u ¥ 0 & (R
s ¥s § s ¡
"6¨¥ ¢ [5] (d) Is
a valid probability space? Be sure to tell me all of the conditions that you
checked to arrive at your answer.
s ¥s § s ¡
!6t¨¥ #¢ [5] (e) Your boss asks you to use the description of
observed. How do you respond? to ﬁnd the probability that a “3” is 2. Tell whether the following statements are “True” or “False”. If you answer “True”, prove the result.
If you answer “False”, give a counterexample. and , then the events C and are independent. are mutually exclusive (i.e. disjoint), then it must be the case that
are independent, then the events d and (R G
SD©
$ [5] (d) If the two events and , then the events A GR G
Sp©
$ [10] (c) If the two events (R G
S#4
$
[5] (b) If the events C and D are independent with
and D cannot be disjoint (i.e. mutually exclusive). and (R G
SY©
$ [5] (a) If the events A and B are independent with
and B cannot be disjoint (i.e. mutually exclusive). are mutually exclusive (i.e. disjoint). must be . ¢ must be (R
SG §¨
©¦ © ¢
£ (R
SG ¢
£ (R
SG , then © ¦ © 4 , if we know that , then ¡ ¥ ¤ ¡
§ and , if we know that ¦ [5] (f) For events and ¢ [5] (e) For events (R
SG d and . 3. A salesman visits one of three cities: Xville, Yville, and Zville. When he visits a given city, the
corresponding probability density function (pdf) of the money that he obtains is given by:
G
¦(
¢4¢
9@9G
G
)¦(
!)
1
! # !
& ( T %9$" G
)¦( ( 6
¥G(7
S¦c8(
)
0 ( 0
E ( T 0
(
0
$
E $ )'
(
42
5 3 $ ¥
SG otherwise
3 [8] (a) Suppose he chooses a city at random to visit. Find the probability that he makes greater than
or equal to $5.
[8] (b) Suppose he chooses a city at random to visit. Given that he makes greater than or equal to $5,
ﬁnd the probability that he visited city .
h [8] (c) Any of the cities can claim that they are the “best” city for the salesman to obtain money  if
they use the correct argument. Give the argument that each can make. In other words, for each city,
give a measure by which it is the “best”.
[6] (d) Suppose he does 20 visits to city , and the money obtained for each visit is indepedent of any
other visit. Write an expression for the probability that he makes more than $150. (Your expression
should only contain simple terms that are easily evaluated.) A
35 2
4C )¥
B )
H"G$ has a cumulative distribution function (CDF) of the form:
0
f
¢©e1!
d0
!¢
T
0
Y 0 D(¥
ESG 4
F 35 2 h C ¥
SG ¥ ¨
b
QS
¥(
b ` X V IQ R
Yc aYWU(
P$
and that make this a valid cumulative distribution function (CDF).
g [5] (b) Find the probability density function
E [5] (a) Find the values of . D
E 5. A continuous random variable i (¥
3 SG 4. [15] I draw an ordered pair
at random (i.e. all points equally likely) from the unit square
, and deﬁne the random variable
. Find the cumulative distribution function
and probability density function
. ...
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 Fall '10
 DennisGoeckel

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