CME308: Assignment 1
Due: Apr. 19, 2012
Part A. Proof of Some Useful Propositions
Problem 1 (10 pts): This problem will cover some useful convergence results.
If cfw_Xn are a series of random variables, we have the following denition.
i=1
p
We say Xn co

CME308: Solutions for Assignment 4. May 26, 2015
Problem 1: Random walk on a graph. Consider a simple random walk on the graph shown in Figure
1. It is the Markov chain which at each time moves to an adjacent vertex, each adjacent vertex having the
same p

CME308: Assignment 4
Due: Tuesday, May 18, 2010
Due Date: This assignment is due on Tuesday, May 18, 2010, by 5pm under the door of 380-383V. See A the course website for the policy on incentives for L TEX solutions. Topics: Markov Chains and First Transi

CME308-2014: Assignment 4
Due: Friday May 30, 5:00pm
Problem 1 (10 pts): Consider a nite state Markov chain with transition probabilities P (x, y), x, y S
and let Fn (x, y) be the probability that y is reached for the rst time at n, starting from x
Fn (x,

CME 298/308: Assignment 3
Due: 5:00 pm, Friday April 29
Do any five problems. (Note that there is some overlap between problems)
They are due Friday April 29, 5:00 pm.
Submit the homework in the bin set up for 298/308 at ICME in the Huang basement. You ca

CME298/308: Assignment 2
Due: 19 April 2016
Do any five problems.
They are due Tuesday April 19, 5:00 pm.
Submit the homework in the bin set up for 298/308 at ICME in the Huang basement. You can also give it
to the instructor (put it in the mailbox on the

CME308/298: Assignment 1
Due: Apr 8, 2016
DO ANY 5 PROBLEMS.
They are due Friday April 8, 5:00 pm.
Submit the homework in the bin set up for 298/308 at ICME in the Huang basement. You can also give it
to the instructor (put it in the mailbox on the first

CME 308 Spring 2016 Notes
George Papanicolaou
June 7, 2016
Contents
1 Sums of independent identically distributed random
1.1 The weak law of large numbers . . . . . . . . . . . . .
1.2 The strong law of large numbers . . . . . . . . . . . .
1.3 Weak conve

CME298-308-2016: Assignment 5
Due: Friday June 3, 5:00pm
Do any five problems.
They are due Friday June 3, 5:00 pm.
Submit the homework in the bin set up for 298/308 at ICME in the Huang basement. You can also give it
to the instructor (put it in the mail

CME298-308-2016: Assignment 4
Due: Tuesday May 17, 5:00pm
Do any five problems.
They are due Tuesday May 17, 5:00 pm.
Submit the homework in the bin set up for 298/308 at ICME in the Huang basement. You can also give it
to the instructor (put it in the ma

CME 308 Spring 2014 Notes
George Papanicolaou
June 4, 2014
Contents
1 Sums of independent identically distributed random
1.1 The weak law of large numbers . . . . . . . . . . . . .
1.2 The strong law of large numbers . . . . . . . . . . . .
1.3 Weak conve

CME308: Assignment 3
Due: 5:00 pm, May 9
Problem 1: In Monte Carlo method, we generate X1 , . . . , Xn with density p(x) and use
an estimate for the quantity Ep f (X) = R f (x)p(x)dx.
1
n
n
i=1
f (Xi ) as
To reduce variance we can use the importance sampl

CME308: Assignment 1
Due: Apr 18, 2014
Problem 1: Some useful results.
This problem will cover some useful convergence results.
If cfw_Xn are a series of random variables, we have the following denition.
i=1
p
We say Xn converges to X in probability, de

CME308: Assignment 2- Solution
Problem 1: Suppose that X1 , . . . , Xn are i.i.d. uniformly distributed in the interval [0, ]. Here is the
unknown parameter to be estimated from the sample.
1. Find the MLE of , denoted by .
2. Prove that converges to in p

CME 308: Practice Problems for the Midterm
Problem 1
Suppose that we wish to generate random variables that are gamma distributed with shape parameter
and scale parameter 1.
a). Prove that the sum of m iid exponential rvs with mean 1 is gamma distributed

Chapter 7
State Space Models and the Kalman
Filter
7.1
State Space Models
We say that an IRd -valued stochastic sequence X = (Xn : n 0) is a state space model if it evolves according
to a recursion of the form
Xk+1 = Fk Xk + Wk + uk
for k 0, where (Fk : k

Chapter 6
Gaussian Random Variables
6.1
Random Variables in IRd
For x IRd , we choose to write x as a d 1 column vector. Let k k2 be the Euclidean norm on IRd given by
v
u d
uX
kxk2 = t
x2i
i=1
Suppose Z is an IRd -valued random vector satisfying E [k] Zk

CME308: Assignment 3 Solutions
Due: 5:00 pm, May 4
Problem 1: In Monte Carlo method, we generate X1 , . . . , Xn with density p(x) and use
an estimate for the quantity Ep f (X) = R f (x)p(x)dx.
1
n
n
i=1
f (Xi ) as
To reduce variance we can use the import

CME308-2015: Assignment 5
Due: Friday June 5, 5:00pm
Do any ve problems
Problem 1: Page rank. The page-rank algorithm computes the equilibrium distribution of a nite state
Markov chain having a transition matrix of the form
P = + (1 )Q,
0 < < 1,
where is

CME308-2015: Assignment 4
Due: Friday May 22, 5:00pm
Do any ve problems
Problem 1: Random walk on a graph. Consider a simple random walk on the graph shown in Figure
1. It is the Markov chain which at each time moves to an adjacent vertex, each adjacent v

CME308: Assignment 3
Due: 5:00 pm, May 5
Do any three problems. (Note that there is some overlap between problems)
Problem 1: In Monte Carlo method, we generate X1 , . . . , Xn with density p(x) and use
an estimate for the quantity Ep f (X) = R f (x)p(x)d

CME308: Assignment 2- Solution
Problem 1: Linear Discriminant Analysis
Linear discriminant analysis, also called Gaussian discriminant analysis, is a very classic generative learning
algorithm in machine learning. In this model,suppose x Rd is the feature

CME308: Assignment 1
Joongyeub & Peng
Problem 1: On the convergence of sequences of random variables.
This problem covers some useful convergence results. In particular implications among various types of
convergence.
If cfw_Xn are a sequence of random v

CME308: Assignment 1
Due: Apr 10, 2015
DO ANY 5 PROBLEMS.
They are due Friday April 10, 5:00 pm. Submit the homework to the instructor (put it in the mailbox on
the rst oor in the math building, 380, or slip it under the oce door, 383V), email it, or hand

CME308: Assignment 2
Due: 24 April 2015
Do any ve problems.
You can give the homeworks to me in class, you can put them in my mailbox in the math department (by
the elevator, rst oor), or slip them under my door 380-383V. You can also email me or the CAs

CME 308 Spring 2014 Notes
George Papanicolaou
June 3, 2015
Contents
1 Sums of independent identically distributed random
1.1 The weak law of large numbers . . . . . . . . . . . . .
1.2 The strong law of large numbers . . . . . . . . . . . .
1.3 Weak conve

CME 308: Assignment 2
Due: Apr 28, 2017
Problem 1
Suppose that we wish to compute E[h(X)], where h is monotone. By use of inversion, we can re-express
this expectation as E[k(U )], where U is uniform on [0, 1] and k is still monotone.
1. Prove that if E[k

CME308: Assignment 3
Due: 5:00 pm, May 9
Problem 1: In Monte Carlo method, we generate X1 , . . . , Xn with density p(x) and use
R
an estimate for the quantity Ep f (X) = R f (x)p(x)dx.
1
n
Pn
i=1
f (Xi ) as
To reduce variance we can use the importance sa

CME308: Assignment 1
Due: Apr 18, 2014
Problem 1: Some useful results.
This problem will cover some useful convergence results.
If cfw_Xn 1 are a series of random variables, we have the following denition.
i=1
p
We say Xn converges to X1 in probability,

CME308: What this course is about
Peter Glynn ([email protected]) A L TEX set by Nick West April 8, 2007
This course is a Ph.D. level introduction to the key ideas and concepts related to the mathematical study of uncertainty. This area is known as