Solutions Question 1 Yes. Any function of data is a statistic. Question 2 Given a random sample 1 , ., from a population distribution with mean 2 : First note that a random sample means the components are all independent. Dene the statistic = ( 1 + 2 )/2
% Code for Problem 1, Assignment week 2, FRE 6083
% Ariola Barci
n=10000;
0umber of steps
x=zeros(n,1);
for i=2:n;
%random walk process
if randn>0
ymmetric RW prob=.5
x(i,1)=x(i-1)+1;
else
x(i,1)=x(i-1)-1;
end
end
s=1:n;
fig1=figure
plot(s,x)
title ('Symm
FRE 6083, Lecture #1
Probability Spaces &Random Variables
(Ross Ch. 1 & 2)
Prof. A. Papanicolaou
Outline of Lecture
Basic review of set-based probability theory,
i.e. outcomes and probabiliNes of events;
mut
Using Random Numbers
Modeling and Simulation of Biological Systems 21-366B
Lecture 2-3
A textbook on probability:
G.R. Grimmett and D.R. Stirzaker
Probability and Random Processes
OXFORD
Motivation
Why randomness?
It seems to be present everywhere.
It may
Quantitative methods Fall 15
Assignment week 1
Andrew Papanicolaou
September 7, 2015
From Ross (5 points each):
Chapter 1:
#1.) A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an
experiment that consists of taking one marble and then replac
FRE 6083, Lecture #3
Markov Chains (M.C.s)
(Ross Ch. 4)
Prof. A. Papanicolaou
Outline of this Lecture
Our discussion of M.C. is moHvated by examples
from insurance and nance, e.g. Markov-
modulated or regime-swi
FRE 6083, Lecture #2
Convergence Concepts, Markov
Processes, and the mar@ngale property
(Ross Ch. 2 and 4)
Prof. A. Papanicolaou
Example: Innite Coin Flips
Let be the probability space of innite binary
seque
Quantitative methods Fall 15
Assignment week 2
Andrew Papanicolaou
September 9, 2015
From Ross Chapter 3:
#8.) An unbiased die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a six and a ve.
1. (10 points)
Quantitative methods Fall 15
Assignment week 1
Andrew Papanicolaou
August 31, 2015
From Ross (5 points each):
Chapter #1: 1, 3, 10,
Chapter #2: 4, 5, 6.
Problem 2 (40 points) This problem focuses on the aggregate loss model
in Insurance in the rst weeks n
FRE 6083, Lecture #4
Stochas7c Processes
(Ross Ch. 4, 5 and 6)
Prof. A. Papanicolaou
Outline of Lecture
Formalize things seen so far into so-called
stochas'c processes.
E.g. random walk, Bernoulli process, Poi