This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2009
Midterm Exam #1
October 22nd, 6:008:00pm, ELAB 303
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. Hint: You may ﬁnd the following fact useful as you solve this exam: , of course, and is i
F g T i sS
i cpS
F g T 8hg g cp
iS
cp g T 8h"D S cS T
$rc e de b !
$ c c T
$ S c
fc e dT $
HF i t
$it
t
Ya
pq XV7UG ES7
b WY ( $ $ T
b` b wxvt
u
g pc
i c ¨ [8] (b) What is is restricted to where is a constant. Let
[7] (a) What is , where Q
$ R $F
HG DE
I
$ CPHG ED7
I$F $
7$CB!A @987 5432§1)'&%#" ©§¥
6 0 ¥ ( $ ! ¨ ¦ ¢
¢¤
£¡ 1. Consider the probability space
deﬁned by: be the outcome of the experiment. ? Your answer should be a number.
? Be sure to derive everything from ﬁrst principles! [5] (c) Since
, it can be treated as a random variable. Find and roughly sketch the cumulative
distribution function (CDF)
and probability density function
of .
. Find the probability space c $ p
eD d$ c (
D $ c p
(
D cp
( $
D $
(
¨ ¨ [5] (a) Find and be the event that dart p is independent of
F c t (
$ y
2. You play a carnival game that consists of throwing three darts at a target. Let
hits the target. Suppose you have the following information: $ 87
be deﬁned as t $ H
[10] (d) Let the random variable
for . .
. , the probability that the third dart hits the target. [5] (b) Given that the third dart hits the target, what is the probability that the second dart hit the target.
[5] (c) Find the probability that all three darts hit the target.
[5] (d) Find the probability that exactly two darts hit the target. has probability density function: ©
else D 3. An exponential random variable with parameter ¡ §¦¤¢
¨£¥£ (
d$ D Suppose that I have three lightbulbs of varying qualities: . . © (
t V$
© !
t $ © ¨ © t c p is said to be memoryless if
( c p is an exponential random variable with hours, where is an exponential random variable with . F ( t ¨ Deﬁnition: A random variable
for all and . ¨ The third dies out after hours, where is an exponential random variable with The second dies out after hours, where ( The ﬁrst dies out after $ [5] (a) Suppose that I use the ﬁrst lightbulb (which lasts a random time ) in my lamp, and it has
lasted two hours. Given such, what is the chance it lasts more than four hours? ¨
[5] (b) Is memoryless? [7] (c) I pick a lightbulb at random (i.e. each of the three is equally likely) and put it into my lamp.
Denote the time that the lamp is lit as . What is the probability that the lamp is lit for more than 2
hours? [8] (d) I pick a lightbulb at random (i.e. each of the three is equally likely) and put it into my lamp.
Denote the time that the lamp is lit as . The lamp is lit for two hours. Given such, what is the
probability that it is lit for more than four hours? memoryless? Explain any difference from (b). ¡ b
(
$ c
© ct
. ¨ #!
$t " # %
F ! t &! c t "
! [10] (d) Find D
g g
b . [5] (c) Find the probability that is given by otherwise [5] (a) Find the value of the constant .
[5] (b) Find t 4. The probability density function of a random variable . $
[5] (e) Is , where: 5. 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 variable with cumulative distribution function
as given below. If you decide to travel to Yville, the proﬁt (in dollars) for the trip is a random
variable with cumulative distribution function
as given below. t
$ ©y
$ y
¡ $ y
$ y ¡ 1 2 3 1 1 1 2 [7] (a) To which village would you travel to maximize your proﬁt?
¥ £
¤ . Find the cumulative distribution function $ y c !c t (
¢ [8] (b) Let . 3 1 ...
View
Full
Document
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

Click to edit the document details