STAT 302
Chapter 1
Combinatorial Analysis
Instructor: Natalia Nolde
Spring 2016
1
A starting example: coin tossing
Consider the following random experiment: tossing a fair coin twice
There are four possible outcomes, namely cfw_HH, HT, TH, TT
What is t
STAT 302
Chapter 5
Continuous Random Variables
Instructor: Natalia Nolde
Spring 2016
1
Introduction
Recall that discrete random variables take values on a finite or a
countable set of numbers (most commonly, integers)
It is, however, possible for a rand
STAT 302
Chapter 4
Random Variables
Instructor: Natalia Nolde
Spring 2016
1
Introduction
A random variable is a real-valued function defined on the sample space
D We use uppercase letters to denote random variables
Example: Toss a fair coin three times
STAT 302
Chapter 3
Conditional probability and Independence
Instructor: Natalia Nolde
Spring 2016
1
Conditional Probabilities: introductory example
Example
Recall a random experiment of tossing a fair coin twice:
S = cfw_HH, HT , TH, TT
D P(tossing two
STAT 302
Chapter 6
Jointly Distributed Random Variables
Instructor: Natalia Nolde
Spring 2016
1
It is often of interest to have a probabilistic model that describes
stochastic behaviour of two or more random variables simultaneously
We begin with the bi
STAT 302
Chapter 8
Limit Theorems
Instructor: Natalia Nolde
Spring 2016
1
Markovs Inequality
Let X be a random variable that takes on only non-negative values.
Then for any real constant a > 0,
P (X a)
E(X)
.
a
Proof:
Define Y as the indicator function
STAT 302
Chapter 7
Properties of Expectation
Instructor: Natalia Nolde
Spring 2016
1
Expected value of random variables
Let X be a random variable. The expectation E(X) gives a measure of
the centre of the distribution of X.
Consider now two random vari
Final exam information
Thursday April 21, 12-2:30pm in OSBO A
Coverage: Chapters 1 - 8 (the exam is cumulative)
The exam will be 2.5 hours, closed book
The standard normal table and the table of moment generating
functions will be provided
You can br
Midterm exam information
Tuesday March 1 (50 min)
Coverage: Chapters 1 - 5 (up to the Common Continuous
Distributions, p.16)
A copy of the formula sheet and all other details are available on the
website
Office hours
D Today 3:30-4:30pm in ESB 3156
D
STAT 302
Chapter 2
Axioms of Probability
Instructor: Natalia Nolde
Spring 2016
1
Sample Space
Definition: A random experiment is an experiment whose outcomes
cannot be predicted with certainty
Definition: The sample space S of a random experiment is the
STAT 302
(Section 202)
Introduction to Probability Theory
It is remarkable that a science which began with the consideration of
games of chance should have become the most important object of
human knowledge.
P.-S. Laplace
Theorie Analytique des Probabili
Winter 2016 STAT 302
Chapter 5 Continuous Random Variables
Learning outcomes:
Aim: Demonstrate an understanding of the basic concepts of continuous
random variables and a number of common continuous distributions
Identify the random variable(s) of intere
Dr. K. Karuppasamy
www.drkk.in
Yahoo Answer dated 07-10-2013
Problem : A fair die is rolled once, and the number score is noted. Let the random variable X be
twice this score, and define the variable Y to be one if an odd number appeared and three if an e
WINTER 2016/17 TERM 2 STAT 302: ASSIGNMENT 1
Due: 2pm on Thursday January 26, 2017
Please remember to include a cover sheet when you submit your assignment. You may hand
in your assignment in class or deposit it in the STAT 302 assignment box on the grou
FALL 2016/17 TERM 2 STAT 302: ASSIGNMENT 3
Due: 2pm on Thursday March 16, 2017
Please remember to include a cover sheet when you submit your assignment. You may hand
in your assignment in class or deposit it in the STAT 302 assignment box on the ground f
STAT 443: Time Series and Forecasting
Chapter 1
Exploratory techniques in time series
analysis
1
What is a time series?
a collection of observations recorded sequentially in time
Abbreviation: time series = TS
2
Examples of time series data
Annual mean of
STAT 443: Course Aims and Objectives
Learning outcomes
The numbered items below each state a learning aim for the course, and
the items that follow indicate the learning outcomes (or objectives) through
which that aim could be deemed to have been satisfie
Activity Solution: Fourier Transforms
We have defined the Fourier transform (FT) of the function h (t) of the
real variable t to be
H () =
h (t) eit dt.
This transform is finite if
|h (t)| dt < .
The inverse Fourier transform (inv. FT) is given by
1
h
Activity Solution: YuleWalker Equations
The i.i.d. sequence cfw_Zt has mean zero and variance 2 . Suppose we
define the stochastic process cfw_Xt by
Xt = 1.30Xt1 0.22Xt2 0.10Xt3 + Zt .
Assume that this process is stationary.
1. How many equations compri
Activity Solution: Model Fitting
The first 200 terms of a time series gave the following results:
k
acf rk
pacf
kk
1
0.80
0.80
2
3
0.67 0.52
0.085 0.112
4
0.39
0.046
5
0.31
0.061
The mean of the observed series was x = 0.03, and c0 = 3.34.
1. What type
Strategies for Effective Studying
I would cram just before the exam and continued to do that because I didnt know how to change and I
should have talked to some people and didnt seek guidance so I got bad marks in first semester. In
second semester I ligh
Activity Solution: Examples of Spectral Densities
The spectral density function f () of a stationary stochastic process X (t)
is defined as
1
(0) + 2
f () =
(k) cos (k)
k=1
for (0, ) , where (k) is the acvf. of X (t) . Basically, f () is the FT of
(k)
STAT 443: Time Series and Forecasting
Chapter 3
Estimation and Model Fitting
for Time Series
Statistics is the grammar of science.
/Karl Pearson/
1
Overview
Given the class of stochastic models introduced in Chapter 2, how to decide
which of the models wo
Activity Solution: BoxJenkins Forecasting
This activity aims to help you understand how to apply BoxJenkins
forecasting methods once a model has been fitted to a time series. The
following model
Xt = 0.5Xt1 + Zt 0.8Zt1 + 0.4Zt2
has been fitted to a series
Activity: Trigonometry Revision
This activity helps refresh you on properties of trigonometric functions,
notably sin and cos. Recall that for any angles A and B we have the following:
sin2 (A) + cos2 (A) = 1,
sin (A + B) = sin (A) cos (B) + cos (A) sin (
y
im
in
ar
STAT406- Methods of
Statistical Learning
Lecture 24
Pr
el
Matias Salibian-Barrera
UBC - Sep / Dec 2016
c Matias Salibian-Barrera, 2016. All rights reserved. Cannot be copied, re-used, or edited.
1
ar
Dimension reduction
y
Principal components a
STAT406- Methods of
Statistical Learning
Lecture 21
Matias Salibian-Barrera
UBC - Sep / Dec 2016
c Matias Salibian-Barrera, 2016. All rights reserved. Cannot be copied, re-used, or edited.
1
Seminar
Date: Thursday, November 24, 2016
Time: 4pm - 5pm
Locati