ECE 603 - Probability and Random Processes, Fall 2002

Final Exam

December 17th, 10:30am-12:30pm, Paige 202

Overview

� The exam consists of six problems for 120 points. The points for each partof each problem are given in brackets - you should spend your two hoursaccordingly.

� The exam is closed book, but you are allowed three page-sides 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 butalso maximize the partial credit should you stumble or get stuck. If you getstuck, attempt to neatly define your approach to the problem and why youare 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 an-swer in terms of the quantities of the previous part that you are unable toobtain.

� Start each problem on a new page. Not only will this facilitate grading butalso 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 bewrong, be sure to write “this must be wrong because . . . ” so that I will knowyou recognized such a fact.

� Academic dishonesty will be dealt with harshly - the minimum penalty willbe an “F” for the course.

Some potentially useful information

����������� ������������������������� ����� � � � � ���"! � ���� ������#��$&%('�)� �����#��$*+�����,�-'. %/�����,��$0+����� �-'.��������$&%('�)� ��������$*+�������-'. 1/����� ��$0+����� �-'.��������$0+�����2�-'.)� �43 �����,��$ ! '. � ��������$ � '.65�����#��$0+���7�8�-'.)� � 3 �����,��$ ! '. ! ��������$ � '.65�����#��$0+�����2�-'.)� �43 ���7� ��$ ! '. � �����#��$ � '.65

Time Function Fourier Transform9 ��$: � '. ; <=;?> �0@< �A� � �CB8DE @F �G:�H�JILK M : M0NOK=P2QR M : M0SOK=P2Q T �-U H�JVXWZY,[ B @]\B @ � sinc �-U �����,� Q2^ U2_�:` �a3Zb �-U ! U2_] � b �-U � U2_]65���7� � Q2^ U2_�:` � � b �-U ! U2_] ! � � b �-U � U2_]

� � <=; cd;?e $ S R � <<gf`h [ �CB @]\ f9 �G:��� I K ! M : M M : M0NiKR M : M0SiK > �-U H� sinc� �-U

Parseval’s Relation: If > �-U is the Fourier Transform of 9 �G:` ,jlk� k M 9 �G:� M �]m :�� jlk� k M > �-U M �.m U

1. [15] Two points are drawn independently and at random from the interval3 R e Q 5 . The outcome (or observation) for the experiment is the distance be-tween the two points. Define a non-trivial probability space for this exper-iment; that is, find ��� e�� e T , where � is the observation space,

�is a set

of subsets of � to which probabilities are assigned, and T is a probabilitymapping from

�to 3 R e K 5 .

2. Random variables > and � have joint probability density function:

U���� a� 9 e� �� I�� e ! K&N 9 N K e R N N 9 � eR e

otherwise

where � is a constant.

[5] (a) Find c.

[5] (b) Find U � � 9 , the marginal probability density function of > .

[5] (d) Find T � > S � , the probability that > is greater than � .

[10] (c) Find U � ; � 9 M , the conditional probability density function for >given � �

(be sure to give limits!).

3. You are measuring a random variable > in a field experiment and find that� 3 > 5 � Q and� 3 > � 5 ��� .

[8] (a) Suppose that the random variable > is input to your system, whichwill malfunction if > � K]Q . What can you say about T � > � K]Q , theprobability that > is greater than or equal to 12?

[7] (b) Repeat part (a), but now assume you have one additional piece ofinformation: you know that > is Gaussian.

4. [15] The random process > ��� is generated as follows: I flip a fair coinrepeatedly. If the first � flips are “heads”, I let > ��� � Q�� ; however, ifany one of the flips before time � is a “tail”, > ��� � R

. (Another way todescribe the same experiment: I flip a fair coin repeatedly as long as I get“heads” and record > ��� H� Q�� for time � ; however, as soon as I get the first“tail”, I then let > ��� 4� R

for all � after that time). Does this sequence ofrandom variables converge? If so, in what ways and to what limiting randomvariable do they converge?

5. Recall that the pulse function F �G:� is defined by:

F �G:�H� ILK e : N �R eelse

[8] (a) A friend tells you that he/she has a wide-sense stationary randomprocess > �G:� that is perfectly correlated over short intervals but then decor-relates; in fact, he/she claims that � ����� � F ��� . Can such a process exist?(Either give an example of such a process or show that such a process cannotexist).

[15] (b) Let > �G:` be a zero-mean wide-sense stationary random process withautocorrelation function � � ��� �� � ��� ; ��; , and let � �G:�H��� c_ > ��2 m � .

� Find the autocorrelation function � a�G: e : � � � 3 � �G: � �G: � 65 . Notethat this gets complicated, but carry it as far as possible. Some of youwill be able to carry it the whole way; if not, indicate how you wouldproceed from where you stopped.

� Is the process � �G:` wide-sense stationary? Be sure to fully justify youranswer.

[12] (b) Let > �G:` be a zero-mean wide-sense stationary random process withautocorrelation function � � ��� �� � ��� ; ��; , and let � �G:`�� _`_ � cc � _`_ > ��2 m � .

� Is � �G:� wide-sense stationary? If so, find its power spectral density a�-U . If not, find the autocorrelation function � 4�G: e : � �� � 3 � �G: � �G: � 65 .� Estimate the power in � �G:` . (Note: Estimate does not mean guess.

There are multiple correct answers, but you must fully justify your an-swer.)

6. Note that you do not have to know anything about estimation to solvethis problem!

[15] The maximum-likelihood estimator (slightly modified to fit this prob-lem) is given as follows:

�> ��� � � � �� argmax � T � � � M > � 9 We study a Poisson (point) process for which the arrival rate � is unknown.Let � be the number of arrivals during 5 seconds of observing the process.Find the maximum-likelihood (ML) estimate of � given that � � K R . [Besure to fully justify your answer].