UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 8
Spring 2014
Issued: Friday, April 18, 2014
Due: In class Thursday, April 24, 2014
Problem 1. Consider a queue that serves at rate (x) > 0 whe

EE126 Discussion 5
Jrme Thai
eo
March 6, 2014
1
Least Squares Estimate
Note 1 (LSE). We know the joint distribution of the pair of random variables (X, Y ) and we want
to estimate X from the observed value of Y . A standard example for the estimation prob

EE126 Discussion 4
Jrme Thai
eo
February 20, 2014
1
Continuous random variables
Problem 1. Let X be uniformly distributed in [0, 1]. Assume that, given X = x, the random
variable Y is exponentially distributed with rate x + 1.
(a) Calculate E[Y ].
(b) Fin

EE126 Discussion 3
Jrme Thai
eo
February 13, 2014
1
MAP and MLE
Note 1 (MLE). Assume that we want to estimate an unobserved population parameter on the
basis of observations y. Let f be the sampling distribution of y, so that f (y|) is the probability of

EE126 Discussion 2
Jrme Thai
eo
February 5, 2014
1
Markov chains
Problem 1. A mouse is in a maze and tries to escape. We assume that the mouse is a Markov
mouse; i.e., the mouse moves randomly from room to room. Moreover, we assume that the mouse
is equal

EE126 Discussion 1
Jrme Thai
eo
January 30, 2014
1
Basic probability
Note 1. For events A, B, C and in the same sample space, we have
Distribution: A (B C) = (A B) (A C)
Union: P (A B) = P (A) + P (B) P (A B)
Independence: A and B independent i P (A B)

EECS 126: Probability and Random Processes
Solutions to Problem Set 12 (Problem 14)
Problem 14
a. The original solution is correct.
b. Let X = X1 + X2 + . + X10 , then E (X ) = 5, V ar(X ) = 5 . Using Chebychev inequality:
6
P (X 7) = P (X 5 2) P (|X 5| 2

EECS 126: Probability and Random Processes
Solutions to Problem Set 12
Problem 1 Processing Jobs
Consider a single job processor with a queue. Suppose that time is slotted into durations
and jobs arrive at the beginning of the time slot as a Bernoulli pr

EECS 126: Probability and Random Processes
Solutions to Problem Set 11
Problem 1 Finite State Markov Chain
Bob goes to Las Vegas. He does not want to lose a lot of money so decides to gamble with
only $3 and to stop playing if he loses these $3 dollars or

EECS 126: Probability and Random Processes
Solutions to Midterm 2
Problem 3 (b)
Some of the statistics of X and Y are E (X ) = 0,
E (X 2 ) =
0.5
x2 dx = 1/12
0.5
E (Y ) = E (E (Y |X ) = E (X 2 + 0.5) = 7/12
E (XY ) = E (E (XY |Y ) = E (0) = 0
The last equ

EE126 Discussion 6
Jrme Thai
eo
March 13, 2014
1
Jointly Gaussian
Note 1. We say that the variables (X1 , , Xn ) are jointly Gaussian, denoted by X := (X1 , , Xn )
N (, ), if the joint distribution has density:
fx (x) =
For X N (, ), we have:
1
1
expcfw_

EE126 Discussion 7
Jrme Thai
eo
March 19, 2014
1
Hidden Markov Model (HMM)
Denition 1. A HMM is a Markov chain cfw_Xn , n 0 with transition matrix P and initial distribution 0 , together with a state observation model cfw_Yn , n 0 with observation matrix

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 7
Spring 2014
Issued: Thursday, April 10, 2014
Due: In class Thursday, April 17, 2014
Problem 1. Consider the following Markov chain with state

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 6
Spring 2014
Issued: Wednesday, March 19, 2014
Due: In class Tuesday, April 1, 2014
Problem 1. Chapter 7 in the notes, Problem 13.
Problem 2.

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 5
Spring 2014
Issued: Saturday, March 8, 2014
Due: In class Thursday, March 13, 2014
Problem 1. Chapter 7 in the notes, Problem 2.
Problem 2. C

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 4
Spring 2014
Issued: Thursday, February 27, 2014
Due: In class Thursday, March 6, 2014
Problem 1. Chapter 5 in the notes, Problem 15.
Problem

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 3
Spring 2014
Issued: Wednesday, February 5, 2014
Due: In class Thursday, February 13, 2014
Problem 1. Chapter 1 in the notes, Problem 24.
Prob

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 2
Spring 2014
Issued: Wednesday, January 29, 2014
Due: In class Thursday, February 6, 2014
Problem 1. Appendix A in the notes, Problem 13.
Prob

UC Berkeley
Department of Electrical Engineering and Computer Sciences
EECS126: Probability in EECS
Problem Set 1
Spring 2014
Issued: Wednesday, January 22, 2014 Due: In class Thursday, January 30, 2014
Problem 1. Appendix A in the notes, Problem 2.
Probl

EE126 Discussion 9
Jrme Thai
eo
April 16, 2014
Problem 1. Let (N (t), t 0) be a Poisson process of rate . Find
P (N (2) N (1) = 1 | N (3) = 2, N (4) N (1) = 2)
Problem 2. Let cfw_Nt , t 0 be a Poisson process with rate . Let Sn denote the time of the n-th

EE126 Discussion 8
Jrme Thai
eo
April 10, 2014
1
Discrete-time Markov chain
Let cfw_Xn n be an (innite) Markov chain on state space S = cfw_1, 2, 3, .
n
Denition 1. State j is accessible from i if there exists n 0 and Pij > 0. If i is accessible from
j an

EECS 126: Probability and Random Processes
Solutions to Problem Set 10 (mid-term 2)
Note: Please send your score to cchang@eecs.berkeley.edu
Problem 1
a) False.
A counterexample: let X and B be independent random variables, X N (0, 1), B is a
Bernoulli ra

EECS 126: Probability and Random Processes
Solutions to Problem 7 (d)
Since the probability of a robbery attempt being successful is 3 , and since the number of
4
candy bars taken in a successful attempt is equally likely to be 1, 2, or 3, we can view eac