Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Questions for Midterm Exam 1 - Random Processes | ECE 534, Exams of Electrical and Electronics Engineering

University of Illinois - Urbana-ChampaignElectrical and Electronics Engineering

Material Type: Exam; Class: Random Processes; Subject: Electrical and Computer Engr; University: University of Illinois - Urbana-Champaign; Term: Spring 2006;

Typology: Exams

Pre 2010

Uploaded on 03/10/2009

koofers-user-ej5
koofers-user-ej5 🇺🇸

10 documents

1 / 14

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Midterm Exam I
ECE534 Spring 2006
There are a total of five problems Mar. 8, 7:00-8:30 pm
You are allowed one sheet (two pages) of notes; no calculators. Each problem is worth 20 points
Please put your NAME here:
1. (a) Let Xbe a uniformly distributed random variable on [0,1].Find the characteristic function
of X.
(b) Let Y1, Y2,... be a sequence of independent random variables uniformly distributed over
{0,1,2,...,9}.Find the characteristic function of Yi.
(c) Let Xn=Pn
i=1 Yi10−i.Find the characteristic function of Xn.
(d) Does the sequence X1, X2, X3,... converge in distribution? If so, what is the limiting
distribution? Clearly justify your answer.
(e) Does the sequence X1, X2, X3,... converge almost surely. If so, what is the distribution of
the random variable to which it converges? Clearly justify your answer. (Hint: The following fact
may be useful: a sequence of non-decreasing, upper-bounded real numbers has a finite limit.)
1
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe

Partial preview of the text

Download Questions for Midterm Exam 1 - Random Processes | ECE 534 and more Exams Electrical and Electronics Engineering in PDF only on Docsity!

Midterm Exam I

ECE534 Spring 2006 There are a total of five problems Mar. 8, 7:00-8:30 pm You are allowed one sheet (two pages) of notes; no calculators. Each problem is worth 20 points

Please put your NAME here:

  1. (a) Let X be a uniformly distributed random variable on [0, 1]. Find the characteristic function of X. (b) Let Y 1 , Y 2 ,... be a sequence of independent random variables uniformly distributed over { 0 , 1 , 2 ,... , 9 }. Find the characteristic function of Yi. (c) Let Xn =

∑n i=1 Yi^10 −i. Find the characteristic function of Xn. (d) Does the sequence X 1 , X 2 , X 3 ,... converge in distribution? If so, what is the limiting distribution? Clearly justify your answer. (e) Does the sequence X 1 , X 2 , X 3 ,... converge almost surely. If so, what is the distribution of the random variable to which it converges? Clearly justify your answer. (Hint: The following fact may be useful: a sequence of non-decreasing, upper-bounded real numbers has a finite limit.)

  1. Suppose you toss a fair toss coin n times. Let Hn be the number of heads seen in the n tosses and let Tn be the number of tails seen in the n tosses. Using the Chernoff bound, show that the probability P (Hn − Tn > nx) is upper-bounded by ( 1 √ (1 + x)1+x(1 − x)^1 −x

)n .

Assume 0 < x < 1. (Hint: Write Hn − Tn as a sum of n i.i.d. random variables.)

  1. Let X be a random variable with mean 0 and variance 1. We wish to estimate X given n observations. The observations are of the form

Yi = X + Wi,

where Wi are independent random variables with mean 0 and variance σ^2 , which are also indepen- dent of X. (a) Show that the linear MMSE estimate of X is given by

Xˆ =

∑^ n

i=

Yi

n + σ^2

(b) Show that the error covariance is given by

σ^2 n + σ^2

(c) For what joint distribution of X, W 1 , W 2 ,... , is the estimate in part (a) the best MMSE estimate (not just the best linear MMSE estimate)? There may be more than one answer to this question, but you have to provide just one answer.

  1. Suppose that X and W are two identically distributed random variables with finite second moments, and let Y = X + W. Find the linear MMSE estimate of X given Y. Does your answer depend on whether X and W are independent or not? (Hint: Eˆ(Y |Y ) = Eˆ(X|Y ) + Eˆ(W |Y ). No major computations are necessary to solve this problem.)