EE 140 - Stochastic Processes, Detection, and Estimation

Department of Electrical and Computer Engineering

Tufts University Spring 2020

Mid-Term Exam

Directions

• You have from 9AM Friday March 27, 2020 until 9AM Tuesday March 31, 2020 to complete

all parts of this exam.

• The exam is open book, open notes, and open Web. Should you need to implement numerical

solutions, you are free to use Matlab, python, Excel, or whatever your favorite computa-

tional tools may be. The use of a symbolic math package such as Wolfram alpha, Maple,

Mathematica, or the like is also fine should you wish to check or avoid altogether messy

calculations.

• Any resources you use that are external to the space between your ears must be referenced

as part of your solution. All code for any software tools you use should be included with

your solution. If there is even a doubt in your mind about whether to include something, just

include it.

• In the event that you find any question to be ambiguous, please describe all assumptions you

have made in generating your answer to maximize your partial credit.

• All solutions should be emailed to Prof. Miller no later than 9AM Tuesday March 31, 2020.

A single PDF file is required.

• Per the Academic Misconduct Policy of Tufts University, you are reminded that cheating,

plagiarism, inappropriate collaboration, academic dishonesty, research misconduct, and facil-

itating the academic misconduct of another are all prohibited in general and with respect to

this exam specifically. To quote from the Policy:

Academic misconduct can occur with the intent to deceive or by disregarding

proper scholarly procedures. Students are responsible for knowing and using proper

scholarly procedures. Faculty are responsible for communicating any course-specific

scholarly procedures. Intention to deceive may be assessed by the scope and context

of the violation. Disregard of proper scholarly procedure is a violation of this policy.

Minor or accidental instances of failing to follow for proper scholarly procedure may

be addressed with the academic setting. Faculty members are required to report

all suspected academic misconduct.

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Problem 1: Consider the random variable X whose probability density function (PDF) is

given by

fX|A(x|a) =

{

2

a

(

1− x

a

)

0 ≤ x ≤ a

0 else

where a > 0 is a real-valued parameter.

1. (10 points) What is aˆML, the maximum likelihood estimate of a given a single obser-

vation X = x?

2. (10 points) Determine the bias of aˆML and Mean Square Error (MSE); i.e., E[(a −

aˆML)

2].

Now suppose we are provided with the following prior distribution for the parameter A

fA(a) =

{

4(1− 2(a− 1)) 1 < a ≤ 3

2

0 else

.

3. (10 points) Determine AˆMAP , the maximum a posteriori estimate of A given a single

observation X = x.

4. (10 points) Determine the Bayes least square estimate of A given a single observation

X = x.

Suppose now that we have hierarchical prior for A of the form

fA|Λ(a|λ) =

{

2λ(1− λ(a− 1)) 1 < a ≤ 1 + 1

λ

0 else

fΛ(λ) =

{

2 1

2

< λ < 1

0 else

where Λ is independent of X.

5. (20 points) Given X = 2, what is AˆLLSE the linear least squares estimate of A?

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Problem 2: In this problem we consider the binary hypothesis testing problem given N

independent and identically distributed (IID) samples Yn for n = 1, 2, . . . , N for which the

two hypotheses are

H0 :fYn(y) =

{

1 0 ≤ y ≤ 1

0 else

H1 :fYn(y) =

{

α exp

{−α (y − 1

2

)}

y > 1

2

0 else

where α is a scalar greater than zero.

1. For N = 1 consider the test for −∞ < τ <∞

y

H1

R

H0

τ. (1)

(a) (10 points) As a function of the threshold τ , determine the detection and false

alarm probabilities.

(b) (10 points) Using the result from the last part of this problem, plot PD vs. PFA

as τ goes from −∞ to ∞.

(c) (10 points) Explain why the curve obtained above is not the receiver operating

characteristic discussed in class.

2. (10 points) Once more with N = 1, determine the maximum likelihood test, δML(y),

for distinguishing H0 from H1. Full credit requires that you explicitly define Yi =

{y | δML(y) = i} for i = 0 and i = 1.

3. Now suppose that N is large. Define the quantity

MN =

1

N

N∑

i=1

Yi

and consider the test

MN

H1

R

H0

τ. (2)

(a) (10 points) Find a value for α such that the test in (2) is optimal as N →∞.

(b) (10 points) For that value of α, determine the detection and false alarm rates for

this test.

(c) (10 points) A common metric for evaluating the performance of a binary hypothesis

test is the area under the receiver operating characteristic, also known as a the

AUC for “area under the curve.” A perfect test would have an AUC of one.

Determine the smallest N such that the AUC for this problem is greater than

1− 10−6. A numerical solution to this problem is fine.

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