Probability Hw 8
Sunhwa
January 28, 2016
(1) A simplied version of the Google page-rank algorithm assigns
a probability distribution R(1), . . . , R(n) to a collection of n webpages.
The webpages are connected by directed links. For each page i, let
L(i)
Probability I - Homework 6
Sunhwa
2015
(1) For a Markov chain, suppose that state i is transient, and state
i is accessible from state j (meaning that there is some integer m s.t.
pji (m) > 0). Show that pij (n) 0 as n
State i being transient tells us th
Probability Hw 7
Sunhwa
2015
(1) A random walk is dened on the integers cfw_ 0,1,2,3,. . . with
the following transition probabilities
1
1
p00 = , p01 =
2
2
1
pn,n+1 = pn,n1 = ; n = 1, 2, . . .
2
Determine whether the walk is transient or persistent. [Hi
Probability I - Homework 1
Sunhwa
Sep
1) Given three events E, F, G, write formulas for the following
events: only E is true; both E and F but not G; at least two of
the events are true
i Let E, F, G S be events (sets of possible outcomes), then let A be
Probability I - Homework 2
Sunhwa
Sep 2015
10) There are two dice on the table. One is a fair die, the other
is xed so that it always comes up showing an even number. You
pick one die at random and roll it, getting an even number. Find the
probability tha
Probability I - Hw 9
Sunhwa
2015
(1) Let cfw_Xt : t 0 be a CMTC on the state space = cfw_1, 2, 3 with
the following transition rates:
12 = 1, 21 = 10
23 = 1, 32 = 5
and all other rates zero.
(a) Find the stationary distribution i
The Generator of the chai
MTH 7241: Fall 2015: Prof. C. King
Assignment 6
Due date: Thursday, October 29.
Reading: slides on Blackboard Ch. 3: Infinite random sequences, and Ch. 4: Infinite
Markov chains.
Problems:
1). For a Markov chain, suppose that state i is transient, and tha
MTH 7241: Fall 2015: Prof. C. King
Assignment 5
Due date: Thursday, October 15.
Reading: slides on Blackboard Ch. 2: Finite Markov Chains. Also Grinstead and Snell
Chapter 11
Note: the text by Grinstead and Snell is available online (free!) at
http:/www.d
MTH 7241: Fall 2015: Prof. C. King
Assignment 7
Due date: Tuesday, November 10.
Reading: slides on Blackboard Ch. 4: Infinite Markov chains.
Problems:
1). A random walk is defined on the integers cfw_0, 1, 2, 3, . . . with the following transition
probabi
MTH 7241: Fall 2015: Prof. C. King
Assignment 8
Due date: Tuesday, November 17.
Reading: slides on Blackboard Ch. 4: Infinite Markov chains and Ch. 5: Large
Deviations for IID sums.
Problems:
1). A simplified version of the Google page-rank algorithm assi
MTH 7241 (Prof. King): FALL 2015: FINAL
December, 2015
You may use your notes and the texts and handouts for the course. You may not use other resources, either
online or offline. Do not discuss the problems with anybody until you have handed back the exa
MTH 7241: Fall 2016: Prof. C. King
Practice Problems for Test 2
1) Consider a Markov chain on a discrete state space. Suppose that state i has period 2, and that state j is
accessible from state i. Prove the following statement or give a counterexample: s
MTH 7241: Fall 2015: Prof. C. King
Assignment 9
Due date: Thursday, December 10.
Reading: slides on Blackboard Ch. 5a: Branching Processes, Ch. 6: Continuous time
Markov chains.
Problems:
1). Let cfw_Xt : t 0 be a CTMC on the state space = cfw_1, 2, 3 wit
MTH 7241: Fall 2015: Prof. C. King
Assignment 2
Due date: Thursday, September 24.
Reading: slides on Blackboard Ch. 1: Elementary Probability Theory. If needed see
Grinstead and Snell for background material.
Problems:
1) Exercises 10, 11, 12, 13, 16, 17
MTH 7241: Fall 2015: Prof. C. King
Assignment 4
Due date: Thursday, October 8.
Reading: slides on Blackboard Ch. 2: Finite Markov Chains, and Grinstead and Snell.
Note: the text by Grinstead and Snell is available online (free!) at
http:/www.dartmouth.edu
MTH 7241: Fall 2015: Prof. C. King
Assignment 1
Due date: Thursday, September 17.
Reading: slides on Blackboard Ch. 1: Elementary Probability Theory. If needed see Grinstead and Snell
for background material.
Problems:
Exercises 1 9 from the Ch.1: Problem
Probability I - Homework 4
Sunhwa
2015
(26) Randomly distribute r balls in n boxes so that the sample
space consists of nr equally likely elements. Let Nn be the number of
empty boxes. Verify that (1 n1 )r is the probability that the i th
box is empty. Su
Probability I - Homework 5
Sunhwa
January 28, 2016
(1) A transition matrix is doubly stochastic if each column sum is
1. Find the stationary distribution for a doubly stochastic chain with
M states.
Let i 1, 2, . . . , M and, j 1, 2, . . . , M , ij be the
Ch. 1: Elementary Probability Theory
Christopher King, Northeastern University
August 4, 2016
0.1
Basics: sample space, events, probability law
Probability theory provides the tools to organize our thinking about how to answer
questions involving randomne
Probability Theory: Ch 1 Problems
September 6, 2016
Exercise 1 Given three events E, F, G, write formulas for the following events: only E is true;
both E and F but not G; at least two of the events are true.
Exercise 2 Two events E and F ; the probabilit
Engineering Probability and Statistics, IE6200
Fall 2016
Class Room: SH 315
Instructor: Professor Fard, Office: 317 SN, E-mail: [email protected]
Prerequisites: Differential and Integral Calculus
Office Hours: Monday, Thursday 10:00-11:00 or by appointment
T
Engineering Probability and
Statistics, IEM G200
By: Professor Nasser Fard
Lecture Six
12/09/16
1
Outline
Continuous Random Variables
Exponential Distribution
Gamma Distribution
Beta Distribution
12/09/16
2
Probability Density Function (p.d.f)
Definition
Engineering Probability and
Statistics, IEM G200
By: Professor Nasser Fard
Fall 1014
Lecture
Tenth
12/09/16
1
Outline
Confidence Interval for
Confidence Interval for difference of
two means
Confidence Intervals for Variances
Comparing Variances of Tw
Engineering Probability and
Statistics, IE6200
By: Professor Nasser Fard
Fall 2013
Outline
What is Probability?
Basic Concepts of Probability
Review on Set Theory
Probability Function
Probability Theorems
Counting Rules
Fall 2013
What is Probability?
Pro
Engineering Probability and
Statistics, IE 6200
By: Professor Nasser Fard
Lecture Ten
Outline
Maximum Likelihood Estimation
Sample Size
Test of Statistical Hypotheses
Tests about one proportion
Tests about two proportions
Tests about one mean when varianc
Engineering Probability and
Statistics, IE 6200
By: Professor Nasser Fard
Lecture
Seven
12/09/16
1
Outline
Distributions of Sums of
Independent Random Variables
Central Limit Theorem
Functions of R. V.
Transformations of R. V.
. Moment Generating Func
MTH 7241: Fall 2015: Prof. C. King
Assignment 3
Due date: Thursday, October 1.
Reading: slides on Blackboard Ch. 1: Elementary Probability Theory. If needed see
Grinstead and Snell for background material.
Problems:
1) Exercises 18, 19, 21, 22, 23, 24 fro