UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 8
Fall 2015
Date: Wednesday, October 21, 2015
Problem 1. A two-dimensional Poisson process is a process of randomly occurring
special
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 8
Fall 2015
Issued: Thursday, October 22, 2015
Due: 9:00am, Thursday, October 29, 2015
Problem 1. Consider the continuous time Marko
EE 126: Probability and Random Processes
Fall 2011
Lecture 22 November 22
Lecturer: Prof. Anant Sahai
Scribe: Kevin Shih
This lecture covers:
Review of the AEP
Bernoulli Process
Poisson Process
22.1
Review of the AEP
Recall from 11/15s lecture we deriv
EE 126: Probability and Random Processes
Fall 2011
Lecture 20 November 15
Lecturer: Prof. Anant Sahai
Scribe: Soi Lon, Lei
This lecture covers:
The Weak Law of Large Numbers
99% of the time!
Central Limit Theorem
20.1
Introduction
In this lecture, we a
EE 126: Probability and Random Processes
Fall 2011
Lecture 19 November 10
Lecturer: Prof. Anant Sahai
Cherno Bounds and Transition to CLT
Scribe: Jared Porter
This lecture covers:
Last Time
Actual Probability vs. Approximations
Cherno Bound
K-L Diverg
EE 126: Probability and Random Processes
Lecture 18 November 15
Lecturer: Prof. Anant Sahai
Fall 2011
Scribe: Brian Lin
This lecture covers:
Law of Large Numbers
Intro to Central Limit Theorem
Markov Inequality
Proof of Weak Law of Large Numbers
Law of La
EE 126: Probability and Random Processes
Fall 2011
Lecture 17.5 November 1
Lecturer: GSI Se Yong Park
.5
Scribe: Min Su Chung
This lecture covers:
Minimum Mean Square Error (MMSE)
Linear Least Squares Estimation (LLSE)
17.1
Minimum Mean Square Error (MM
EE 126: Probability and Random Processes
Fall 2011
Lecture 17: 10/27/11
Lecturer: Prof. Anant Sahai
Scribe: Brian Suh
This lecture covers:
General 2D Case
n-dimensional Gaussian
17.1
Recap
In the last lecture, we dened that two random variables X and Y
EE 126: Probability and Random Processes
Fall 2011
Lecture 15 October 20
Lecturer: Prof. Anant Sahai
Scribe: Yin Huang
This lecture covers:
Gaussian random variables continued: Gaussian in 2 dimensions
15.1
Gaussian random variables continued
Recall that
EE 126: Probability and Random Processes
Fall 2011
Lecture 14 October 18
Lecturer: Prof. Anant Sahai
Scribe: Andrew Lee
This lecture covers:
The Gaussian Random Variable
Derivation
Properties
The Standard Normal
14.1
The Gaussian Random Variable
This
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 4
Fall 2015
Issued: Friday, October 2, 2014
Self-graded Scores Due: 5:00pm Monday, October 5, 2015
Submit your self-graded scores via the
EE 126: Probability and Random Processes
Fall 2011
Lecture 23 November 29
Lecturer: Prof. Anant Sahai
Scribe: Lisa Yan
This lecture covers:
Introduction to Markov Chains
A Vector View on Markov Chains
23.1
What are Markov Chains?
So far, the random phen
EE 126: Probability and Random Processes
Fall 2011
Lecture 17.1 Nov 1
Lecturer: TA. Se Yong Park
.1
Scribe: Chao Liu
This is a review lecture, we go over two problems in the exercise exam:
practice midterm 1.1
Prove MMSE
practice midterm 1.2
17.1
pract
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 3
Fall 2015
Issued: Thursday, September 17, 2014
Self-graded Scores Due: 5:00pm Monday, September 21, 2015
Submit your self-graded scores
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 3
Fall 2015
Issued: Thursday, September 10, 2015 Due: 9am, Thursday, September 17, 2015
Problem 1. This problem will explore an impo
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 1
Fall 2015
Issued: Thursday, September 3, 2015
Self-graded Scores Due: 5pm, Monday, September 7, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 2
Fall 2015
Issued: Thursday, September 03, 2015 Due: 9am, Thursday, September 10, 2015
Problem 1. Consider a binary tree with n lev
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 1
Fall 2015
Issued: Thursday, August 27, 2015
Due: 9am, Thursday, September 03, 2015
Problem 1. Find an example of 3 events A, B, an
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 2
Fall 2015
Issued: Thursday, September 10, 2015
Self-graded Scores Due: 5:00pm Monday, September 14, 2015
Submit your self-graded scores
Fall 2009: EECS126 Practice Midterm 2
No Collaboration Permitted. One sheet of notes is permitted. Turn in with your exam.
Be clear and precise in your answers
Write your name and student ID number on every sheet.
Come to the front if you have a question.
Fall 2009: EECS126 Practice Midterm 1
No Collaboration Permitted. One sheet of notes is permitted. Turn in with your exam.
Be clear and precise in your answers
Write your name and student ID number on every sheet.
Come to the front if you have a question.
EE 126: Probability and Random Processes
Fall 2011
Lecture 13 October 13
Lecturer: Prof. Anant Sahai
Scribe: Ziang Xie
This lecture covers:
Moment Generating Function (continued)
Estimation and Iterated Expectations
13.1
Recap
Thus far we have been anal
EE 126: Probability and Random Processes
Fall 2011
Lecture 10 September 29
Lecturer: Prof. Anant Sahai
Scribe: Kenrick Lam
This lecture covers:
Information about the rst midterm
Continuation of the discussion of conditioning on continuous random variabl
EECS 126 Probability and Random Processes
Kannan Ramchandran
University of California, Berkeley: Fall 2014
September 19, 2014
(Practice Version) Midterm Exam 1
Last name
First name
SID
Rules.
DO NOT open the exam until instructed to do so.
Note that the
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution of Sample Midterm
Fall 2014
Problem 1.
(a)
4
3
P (3Q) + P (4K) P (3Q&4K) =
48
7
+
4
4
48
6
52
10
4
4
3
4
44
3
(b)
P (a box is empty) = (
So rrrr,a,us
FaIl 2009: EECS126 Finat
No Gollaborati,an Pennitted,. Two sheets of notes permitted,. Tfu,rn i,n wdth
yaur
e&o,nx.
Be clear and precise in your answerg
your
Write
name and student ID numb,er on every sheet.
Come to the front if you have a qu
EECS 126 Probability and Random Processes University of California, Berkeley: Spring 2011
Abhay Parekh April 12, 2011
Midterm Exam
Last name First name SID
Rules.
0 You have 80 mins (3:40pm 5pm) to complete this exam.
0 The exam is not open book,
Recitation 4
1. (a) Derive the expected value rule for functions of random variables E[g(X)] =
!
x g(x)pX (x).
(b) Derive the property for the mean and variance of a linear function of a random variable
Y = aX + b.
E[Y ] = aE[X] + b,
var(Y ) = a2 var(X).
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 6
Spring 2017
Date: Wednesday, March 1, 2017
Problem 1. Let X N (, 2 ) and Y Poi(). Find the Chernoff bounds for
(a) P (X )
(b) P (Y
UC Berkeley
Department of Electrical Engineering and Computer Sciences
Electrical Engineering 126: Probability and Random Processes
Discussion 9
Spring 2017
1. Illegal U-Turns
Each morning, as you pull out of your driveway, you would like to make a
U-turn
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 9
Spring 2017
Date: Wednesday, April 5, 2017
Problem 1. (Infection source detection) Consider a graph where each node represent
each
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 9
Spring 2017
Date: Wednesday, April 5, 2017
Problem 1.
(a) The MLE of the source is as follows.
u
= arg max P (G|u)
u
Note that the
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 7
Date: Wednesday, March 8, 2017
Problem 1. (Final Sp06) Consider a particle moving according to the following
Markov Chain:
Figure 1