Homework 1
Music 26AC
A DIFFERENT MIRROR (TAKAKI)
1. What incident does Takaki use as a springboard for the essay? Why?
2. What are two (2) examples Takaki gives of how various racial or ethnic groups were
prominent in or impacted U.S. history?
3. What is
Operations Research II
University of California, Berkeley
Spring 2006
Midterm 1
1. The number if storms in the upcoming rainy season is Poisson distributed but with a parameter
value that is uniformly distributed over (0, 5). i.e. is uniformly distributed
IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 7 Suggested Solution Chapter 5. 70. (a) Let N1 (t) be the number of departures by time t, and S be the service time for the rst customer. From example 5.25, we know th
558
8
Queueing Theory
Exercises
1. For the M/M/1 queue, compute (a) the expected number of arrivals during a service period and (b) the probability that no customers arrive during a service period. Hint: Condition. *2. Machines in a factory break down at
Exercises
407
raising that matrix to the nth power (by utilizing k matrix multiplications). It can be shown that the matrix (I Rt/n)1 will have only nonnegative elements.
Remark Both of the preceding computational approaches for approximating
P(t) have pr
IEOR 161 Operations Research II
After a basic review of probability concepts, we will cover conditional probability and expectation,
Markov chains, exponential distribution and Poisson processes. Time permits, renewal the
Lecture 3
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
January 24th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 3
1/13
Uncorrelated is not the same as independent
Uncorrelated random variables are not necessarily independen
IND ENG 173
Introduction to Stochastic Processes
Prof. Mariana Olvera-Cravioto
Assignment #4
February 10, 2017
Page 1 of 2
Assignment #4 - due Friday, February 17th, 2017
1. Consider the Markov chain on the states cfw_1, 2, 3, 4 having one-step transition
Lecture 5
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
January 31st, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 5
1/13
A Markov chain for weather forecast
I
Suppose we want to model the weather in Berkeley as a Markov chain
Lecture 4
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
January 26th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 4
1/12
More on conditional expectation and variance
I
The conditional expectation and the conditional variance
Lecture 1
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
January 17th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 1
1/11
Introduction
I
A stochastic process is a collection of random variables, e.g.,
cfw_Xn , n N
or
cfw_Yt ,
IEOR 161 Homework 3& 4
Due BEFORE Lecture on Thursday, Feb. 11, 2016
Unless specified otherwise, all home works are taken from the textbook,
Introduction to Probability Models, 9th edition, by S. Ross.
Chapter Two, Problems 60, 61, 63, 67, 69, 76, 77
1
IEOR 161 Homework 8
Due BEFORE Lecture on Tuesday, March 29th, 2016
Unless specified otherwise, all home works are taken from the textbook,
Introduction to Probability Models, 9th edition, by S. Ross.
Chapter Four, Problems 18, 20, 26, 29, 31, 33.
1
IEOR 161 Homework 2
Due BEFORE Lecture on Tuesday, Feb. 2nd, 2016
Unless specified otherwise, all home works are taken from the textbook,
Introduction to Probability Models, 9th edition, by S. Ross.
Chapter One, Problems 30, 45;
Chapter Two, Problems 42
Lecture 9
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
February 14th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 9
1/11
Time reversibility
I
We say that the Markov chain cfw_Xn : n 1 is time reversible if its
stationary dis
Lecture 7
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
February 7th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 7
1/10
Stationary distributions
I
Let cfw_Xn : n 0 be a Markov chain having one-step transition matrix P .
I
We
Lecture 2
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
January 19th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 2
1/15
Using the PMF to compute probabilities
Experiment: Toss a fair coin 10 times. What is the probability th
Lecture 8
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
February 9th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 8
1/13
Convergence theorems
I
Theorem: Suppose cfw_Xn : n 0 is irreducible, aperiodic, and has a
stationary dis
Lecture 6
Mariana Olvera-Cravioto
UC Berkeley
[email protected]
February 2nd, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 6
1/12
The hitting time of a state
I
For any set A S, we will write
Px (A) = P (A|X0 = x),
i.e., the probabilit
lE c tcfw_ lb I
*dB
Fx^-plt, ff
A' r of
e
.ol\
Cl.
t\
-Lo
e.*iu<
errds
Jorv\t
rl
Fl
Lo)
l-in ct
T\.
\e
[i,)
F iNJ
rk
\'.
Fr^J
+\'e
t)
cl
)olt'
.bih
!"1 l,+/
'Y^0b0
P
q cccfw_ d
Sud
Yc
i
r
iu1
ercfw_\er
r^
cfw_urn
'cfw_ 6,
,
L,is
oY
he"
t"
obJa;n
5tliv1
^