 # LECTURE NOTES:              SUMMARIES:

Brief Preview
• CHAPTER 1                                                     - WEEK 1
• CHAPTER 2                                                     - WEEK 2
• CHAPTER 3                                                     - WEEK 3
• CHAPTER 4                                                     - WEEK 4                         - REVIEW FOR TEST 1
• CHAPTER 5                                                     - WEEK 5
• CHAPTER 6                                                     - WEEK 6
• CHAPTER 7                                                     - WEEK7
• CHAPTER 8                                                     - WEEK 8                         - REVIEW FOR TEST 2
• CHAPTER 9                                                     - WEEK 9
• CHAPTER 10                                                   - WEEK 10
• CHAPTER 11                                                   - WEEK 11
• CHAPTER 12
•                                                                           -WEEK 12                       - FORMULAS FOR FINAL EXAM
Review for Test 2 (2004)

# MATH 2F05 Applied Advanced Calculus

## CLASS TRANSPARENCIES:

#### DIFFERENTIAL EQUATIONS

Case of multiple eigenvalues (suppliment)
Bessel-equation example

### SECOND MIDTERM

Topics: Special ODEs and their solutions (Bessel functions, etc.). Simplifying ODEs by introducing a new dependent AND independent variable (expect a questin of this type). Applications of dot and cross products. 3D curves and their properties. Scalar and vector fields (grad, div and curl). Type I and Type II (both path dependent and independent) line integrals. Double integrals, polar coordinates. Type I surface integrals. Length, area, volume, center of mass, moment of inertia!

## LABS

LAB 0:  Basics of Maple
LAB 1:  Exploring, graphically,  families of solutions to 1st-order ODE
LAB 2:  Verifying solutions. Solving and visualizing exact equations.
LAB 3    Integrating factor. Clairaut equation. Orthogonal families of curves.
LAB 4    Emptying cylindrical, conical and semi-shperical container - rescaling. The A+B->C  chemical reaction.
LAB 5    Variation-of-Parameters and Undetermined-Coefficients techniques
LAB 6    Gaussian elimination and backward substitution in the context of V of P technique.  Dealing with multiple roots of a polynomial.
LAB 7    Computing eigenvalues by 'gausselim' and eigenvectors by 'gaussjord'. Finding s-vectors and  u-vectors.
LAB 8    Dealing with complex solutions. Solving  y' = A.yby  Undetermined Coefficients and Variation of Parameters.
LAB 9
LAB 10
LAB 11
LAB 12                Velocity, acceleration, curvature and torsion; osculating circle.
LAB 13
LAB 14                Type I and II line integrals
LAB 15                Double integrals; area, CM and MI of planar lamina
LAB 16                Surfaces. Type I and II surface integrals
LAB 17
LAB 18
LAB 19                Complex functions - differentiation and integration
LAB 20
LAB 21

## ASSIGNMENTS AND SOLUTIONS

Assgn. 1                    Sol. 1
Assgn. 2                    Sol. 2
Assgn. 3                    Sol. 3
Assgn. 4                    Sol. 4
Assgn. 5                    Sol. 5
Assgn. 6                    Sol. 6
Assgn. 7                    Sol. 7
Assgn. 8                    Sol. 8
Assgn. 9                    Sol. 9
Assgn. 10                  Sol. 10
Assgn. 11                  Sol. 11            Maple supplement
Assgn. 12                  Sol. 12
Assgn. 13                  Sol. 13
Assgn. 14                  Sol. 14
Assgn. 15                  Sol. 15
Assgn. 16                  Sol .16
Assgn. 17                  Sol. 17
Assgn. 18                  Sol. 18
Assgn. 20                  Sol. 20
Assgn. 21

# MATH 2P81 Probability

## LECTURE NOTES:

Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Normal Approximation

### MAPLE BASICS

FINAL EXAM:    Summary of concepts and formulas        Other Comments

### ASSIGNMENTS AND SOLUTIONS

Each solution will be available only after the corresponding assignment's deadline.

### REVIEW

OLD TEST 1 (includes a solution sheet)
ANOTHER TEST 1                                                 Solution
2007 First Midterm                                                   Solution
2008 First Midterm                                                   Solution
2010 First Midterm                                                   Solution
2010 Substitute Midterm                                           Solution
2011 First Midterm                                                   Solution
2012 First Midterm                                                   Solution
OLD TEST 2 (with solution)
ANOTHER TEST 2                                                  Solution
2007 Second Midterm                                               Solution
2009 Second Midterm                                               Solution                    Maple program
2010 Second Midterm                                               Solution
2011 Second Midterm                                               Solution
2012 Second Midterm                                               Solution
Final Exam 2006                                                        Solution
Final Exam 2007                                                        Solution
Final Exam 2008
Final Exam 2009                                                        Solution
Final Exam 2010                                                        Solution
Final Exam 2011                                                        Solution
Final Exam 2012                                                        Solution

## Introduction to Maple

Maple's treatment of special discrete distributions
Continuous distributions by Maple

# MATH 2P82 Mathematical Statistics

## Introduction to Maple

### Summary of concepts and formulas

Lab 1        (Basic Continuous Distributions: Uniform, Normal, Exponential, Gamma and Cauchy)
Lab 2        (Central Limit Theorem: Adding many independent random variables results in Normal distribution)
Lab 3        (Basic Discrete Distributions: Binomial, Negative Binomial, Poisson, Hypergeometric)
Lab 4        (Special Distributions: Student, Fisher, chi-square; Convolution of two Normal, and of two exponential RVs)
Lab 5        (Piecewise pdf, F(x), mean and varinace, median and SIQR; pdf of an order statistic, of a sample median - approaching Normal)
Lab 6        (The first 3 midterm questions)
Lab 7        (Question 4 of last exam;  Finding, numerically, Maximum-Likelihood estimators of binomial-distribution parameters)
Lab 8        (Question 5 of last exam; The bivariate Normal distribution: joint, marginal and conditional  pdfs, the role of rho)
Lab 9        (Regression: scattergram, fitting the least-squares straight line; Correlation: constructing CI for rho)
Lab 10      (Multivariate regression; One-way analysis of variance)
Lab 11      (Two-way analysis of variance without interaction; with interaction and replicates).

## DATA FOR:

Assignment 8, Question 2
Assignment 9, Questions 1 and 3

Assignment 1
Assignment 2
Assignment 3
Assignment 4
Assignment 5
Assignment 6
Assignment 7
Assignment 8
Assignment 9

# MATH 3P60 Numerical Methods

## TRANSPARENCIES:

General Preview
Interpolating polynomials
Least-square fit (discrete data)
Least-square fit to functions (Legendre polynomials)
LS fit with weights (Chebyshev polynomials)
Numerical integration (trapezoidal rule); Romberg's technique
Simpson's rule
Designing own formulas. Improper integrals
Gaussian integration
Numerical Differentiation
Solving 2nd order ODE with given boundaries; LU decomposition of tri-diagonal matrix
Solving non-linear equations with one or more unknowns
Non-linear ODE with two boundary values
Runge-Kutta technique for solving (a set of) ODEs
Trigonometric (Fourier) approximation; continuous and discrete case
Final review

## LABS

Maple's basics
Interpolating polynomials (Newton and Lagrange technique)
Polynomial least-square discrete fit (to a set of point); weights; linear models
Polynomial LS continuous (ie. to functions) fit; Gram-Schmidt and Legendre polynomials
Improved (weighted) LS fit using Chebyshev polynomials
Gaussian elimination with full pivoting
Gram-Schmidt; composite trapezoidal rule; Romberg algorithm
Simpson's rule
Other formulas. Integrals with a weight function
Gaussian integration
Numerical differentiation
LU decomposition; boundary-value ODE; Richardson exptrapolation
Newton's technique
Non-linear, second-order ODE
Solving ODE of any order - initial value problem
Trigonometric-polynomial fit to various function
Interpolating trigonometric polynomial; Fast Fourier transform
Final review

## ASSIGNMENTS AND SOLUTIONS

Assgn. 1      Assgn. 5                         Sol. 5
Assgn. 6

# MATH 3P82 Regression Analysis

## LECTURE NOTES

### Previous second-midterm solution

Assignment 1         Solution 1       (Maple program)
Assignment 2         Solution 2       (Maple)
Assignment 3         Solution 3       (Maple)
Assignment 4         Solution 4       (Maple)
Assignment 5         Solution 5       (Maple)
Assignment 6         Solution 6       (Maple)
Assignment 7         Solution 7       (Maple)
Assignment 8         Solution 8       (Maple)
Assignment 9         Solution 9       (Maple)
Assignment 10       Solution 10     (Maple)
Assignment 11

### DATA FOR:

Assgn. 1     (contains x and y)
Assgn. 6
Assgn. 7
Assgn. 8     (x and y for Question 1)
Assgn. 9

# MATH 3P85 Mathematical Statistics II

## LECTURE NOTES

### Lecture 5Lecture 6Lecture 23Lecture 11

Assgn. 1
Assgn. 2
Assgn. 3
Assgn. 4
Assgn. 5
Assgn. 6
Assgn. 7
Assgn. 8
Assgn. 9
Assgn. 10

Assignment 1
Assignment 2
Assignment 3
Assignment 4a
Assignment 4b
Assignment 5

# MATH 4P84 - Topics in Stochastic Processes and Models

## Course Outline

### Lab 10    Playing two patterns against each other; game's duration

Assgn. 1          Due date:     Jan. 17
Assgn. 2          Due date:     Jan. 31
Assgn. 3          Due date:     Feb. 14
Assgn. 4          Due date:
Feb. 28
Assgn. 5          Due date:     March 13
Assgn. 6          Due date:     April 3

First Midterm 2013    Maple Solution     2nd midterm 2013    Maple solution     FINAL EXAM 2013      Maple Solution      First Midterm 2014      Maple solution

Second Midterm 2014         Maple solution      FINAL EXAM 2014       Maple Solution             Midterm 1 (2015)      Maple solution      Midterm 2 (2015)     Maple solution     2015 FINAL     Maple solution

First Midterm 2017     Maple Solution      2nd Midterm 2017     Maple Solution       FINAL Exam 2017      Maple Solution           2018 First Midterm     Maple solution    2018 Second midterm    Maple solution

2018 FINAL        First Midterm 2019        Maple Solution      Second Midterm 2019   Maple Solution    2019 Final    Maple solution    2020 First Midterm    Maple solution       2020 Second Midterm    Maple Solution

# MATH 4P85 - Topics in Advanced Statistics

## ASSIGNMENTS  (numerical answers should be given in decimal, to at least 4 significant digits - make sure to click 'reload' to get the latest version)

Assgn. 0        Due: Sept. 16     Maple solution
Assgn. 1        Due: Sept. 27     Maple solution
Assgn. 2        Due: Oct. 11      Solution
Assgn. 3        Due: Oct. 30      Maple solution
Assgn. 4        Due: Nov. 8       Solution
Assgn. 5        Due  Nov. 22     Maple solution
Assgn. 6        Due. Nov. 29     Solution

## OLDER YET LABS   and the corresponding 'legend'

Maple basics; PGF; generating random samples; composite distributions.
Generating Poisson process (two ways). Time till next arrival.
Sampling from Uniform and Poisson distributions. Non-homogeneous PP.
Yule process. Special case of Pure Death process.
Linear-growth process. Simulating a general B&D Markov process. Time till absorption.
Linear Growth with immigration. M/M/infinity queue. Power-supply process.
Stationary probabilities of a B&D process. Probability of, and time till, extinction.
P = exp(A). Eigenvalues and constituent matrices of A. Computing a function of A.
Brownian motion.
Markov, Yule and other Autoregressive models.

## OLD LABS

Examples of PDF, MGF, PGF, RIS and CLT
Total probability formula, marginal and conditional PDF, generating random #s from various distributions
Simulating Poisson process in two different ways; time till the next arrival after 1:00

## EXAMPLES OF:

First Midterm               Solution
Another First Midterm
2006 First Midterm      Solution
2005 First Midterm      Solution
2007 First Midterm      Solution
2008 First Midterm      Solution      Maple workspace
2009 First Midterm      Solution
2011 First Midterm      Solution
2012 First Midterm      Solution       2013 First Midterm            Solution (Maple)     2014 First Midterm         Solution (Maple)          2015 First Midterm         Solution (Maple)       1st midterm 2016 winter    Maple solution
1st midterm 2016 fall    Maple solution
Second Midterm           Solution
2005 Second Midterm  Solution
2006 Second Midterm  Solution
2007 Second Midterm  Solution
2011 Second Midterm    Maple workspace
2012 Second Midterm    Solution                         2013 Second Midterm       Solution       2014 Second Midterm      Solution (Maple)       2015 Second Midterm     Solution (Maple only)    2nd midterm 2016    Maple solution
2016 Fall, Second Midterm          Maple solution
2004 FINAL                  Solution
2005 FINAL                  Solution
2006 FINAL                  Solution
2007 FINAL
2008 FINAL     Maple Solution          2009 FINAL             2010 FINAL         Maple Solution        2011 FINAL   Maple "Solution"    2012 FINAL    Maple "solution"     2013 FINAL      Maple "solution"   2014 FINAL      Maple "solution"
2015 FINAL          Maple "solution"             2016 winter FINAL           Maple solution          2016 fall Final          Maple Solution         2017 First Midterm     Maple Solution      2017 Second Midterm       Solution    2017 Final    Maple Solution
2018 First midterm     Maple Solution      2018 Second midterm       Maple solution       2018 Final exam    Maple solution   2019 First Midterm   Maple Solution    2019 Second Midterm    Maple Solution     2019 Final   Maple solution

# MATH 5P84   Time Series

## 'LABS'

Multivariate Normal distribution, partial correlation coefficient
Distribution of sample correlation coefficient
Solving (through eigenvectors) V = A.At (don't use in your Assgn. 1)

Assgn. 0    Due:  Sept. 11
Assgn. 1    Due:  Sept. 27  9 am
Assgn. 2    Due:  Oct. 4     9 am
Assgn. 3    Due:  Oct. 11
Assgn. 4    Due:  Oct. 25
Assgn. 5    Due:  Nov. 1
Assgn. 6    Due:  Nov. 8         Data (found inside a Maple file to be downloaded)
Assgn. 7    Due:  Nov. 15       Data
Assgn. 8    Due:  Nov. 25   in class
Assgn. 9    Due:  Dec. 3

### 2006 FINAL EXAM

Data:     Q3.m                  Q4.m

Maple 'solution' (to be presented in proper manner)

Data

### 2008 FINAL

DATA:     Q_3.m      Q_4.m      Q_5.m

### 2009 FINAL

Data (always called X):   Q1.m    Q2.m    Q5.m

2010 FINAL           Q4.m           Maple solution

2011 FINAL      Data for Q3, list w          Data for Q4, array y       Maple solution

2012 FINAL  (open book)    Data for exam (sequence X)      Topics: conditional distr., ARMA models, smoothing of spectra, circular Markov, ML estimation.

2013 Final         Data for Question 3 (list W)          Data for Question 4 (list Y)         Data for Question 5 (list X)     Maple solution

## LECTURE NOTES

Introduction to Large-sample Theory
Expanding ML estimator
Extending Central Limit Theorem
Generating random numbers from U(0,1)
Linear congruential generator
Monte Carlo techniques
Approximating g(Xbar) distribution
Beyond Normal Approximation - Introduction

## 'LABS'

Xbar^3*Ybar^2*Zbar expanded into SIPs
Approximation to Gamma and Poisson distributions
Approximating the r distribution
Example of bivariate approximation
Method-of-moments estimators - gamma distribution
ML estimators of parameters of gamma distribution
Monte Carlo simulation of a sample from a distribution

### Maple Examples (updated by MC)

Assgn. 1           Due:   Jan. 17
Assgn. 2           Due:   Jan. 24
Assgn. 3           Due:   Jan. 31
Assgn. 4           Due:   Feb. 7
Assgn. 5           Due:   Feb. 26
Assgn. 6           Due:   March 6
Assgn. 7           Due:   March 13
Assgn. 8           Due:   March 20
Assgn. 9           Due:   March 27
Assgn. 10         Due:   April 3

### 2017 FINALMaple Solution2019 FINALMaple Solution

All key steps and answers must be entered, clearly and legibly, in the examination booklet.

# MISCELLANEA

## Euler-Maclaurin formula and Stirling approximation 