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MATH 1P98 Basic Statistical Methods


Brief Preview
                                                                                                                                Review for Test 2 (2004)

MATH 2F05 Applied Advanced Calculus





Case of multiple eigenvalues (suppliment)
Bessel-equation example



First order differential equations: trivial, separable, (modified) scale independent, linear, Bernoulli, exact (integrating factors), Clairaut, interchanging  x and y, introducing new dependent variable, orthogonal families of curves.

Second order: y missing, x missing, linear (variation of parameters), with constant coefficients (3 cases), non-homogeneous (variation of parameters, undetermined coefficients), superposition principle, Cauchy equation.


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!

2005 Progress Exam                   Maple Solution

First Midterm                               Solution

Second midterm                            Solution

Sample Midterm from 2003          Solution

2003 Progress Exam

2004 Progress Exam               Solution to Questions 1-6       Solution to Questions 7-11 

First-Semester Review

Sample 2nd midterm                and another one     

2005 Second midterm               Maple solution

Substitute 2nd midterm             Solution

Topics covered by the final exam

Sample (1998) Final Exam              Another (1999) Final Exam

2005 Final Exam                      Solution


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
                Gradient, divergence and curl
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  


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. 19                  Sol. 19
Assgn. 20                  Sol. 20
Assgn. 21                  Sol. 21
Assgn. 22                  Sol. 22

MATH 2P81 Probability



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



Maple Basics - PDF version

FINAL EXAM:    Summary of concepts and formulas        Other Comments     


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





Know your distributions

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



SECOND MIDTERM                         AND ITS SOLUTION

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).



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


MATH 2P20 Numerical Analysis I

MATH 3P60 Numerical Methods



Intepolating polynomials (both Newton and Lagrange technique)

Least-square fit of discrete data (a table, which may include weights) by a polynomial, or any other linear model

Gaussian elimination with back substitution (both with exact fractions and decimals - the latter requiring full pivoting)

Least-square fit of a function (in any finite interval) by a polynomial, utilizing Legendre polynomials

Test                   Solution


Fitting a function by a polynomial utilizing Chebyshev polynomials.

Trapezoidal and Simpson's (composite) rules; Romberg's algorithm.

Designing 3- and more-point formulas, without and with a weight function.

Gaussian formulas, without and with a weight function.

Designing formulas for numerical differentiation; Richardson interpolation.

Solving one or more non-linear equations.

Second-order ODE (linear and non-linear) with given boundary values.

Test                Solution


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 


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  


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

MATH 3P82 Regression Analysis



The "missed" lab


Previous first-midterm solution

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


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

Student are allowed to use all basic features of Maple, but NOT any of its 'libraries', such as 'with(Statistics)'

Review for Final Exam


Math Pre-requisites

Lecture 5    Lecture 6   Lecture 23      Lecture 11           

ASSIGNMENTS      Due date:                                 Press reload to get the latest version!

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

MATH 4F83  Statistics


Dec. 2000

Dec. 1999


March 2001

March 2000


April 2000

April 2001


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

MATH 4P84 - Topics in Stochastic Processes and Models  

Course Outline

Lab 1      Maple for beginners + how to use PGF

Lab 2      Short-range forecast, initial distribution, stationary probabilities

Lab 3      Classification of states + when all recurrent states are absorbing

Lab 4      All recurrent states absorbing; probability of, and time till, absorption

Lab 5      Large-power-of-P limits: Case 2, Case3 and Case 4

Lab 6      Gambler's Ruin; Solving difference equations; Polynomial roots

Lab 7      Composition of distributions. Branching Process: Generation m

Lab 8      Branching Process, Part 2: Truncated and Total Progeny

Lab 9      Breaking even; generating a specific pattern of successes/failures

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

ASSIGNMENTS     (press reload to get the latest version)

Assgn. 1          Due date:    
Assgn. 2          Due date:    
Assgn. 3          Due date:    
Assgn. 4          Due date:  
Assgn. 5          Due date:    
Assgn. 6          Due date:    

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

Course Outline

'PREREQUISITES'   with formal proofs   and a table of common distributions

SUMMARY            Solving linear, 1st-order PDEs                Constituent Matrices

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:
Assgn. 1        Due:
Assgn. 2        Due:
Assgn. 3        Due:
Assgn. 4        Due:
Assgn. 5        Due
Assgn. 6        Due. 

2019 LABS 

Lab 1

Lab 2

Lab 3

Lab 4

Lab 5

Lab 6

Lab 7

Lab 8

Lab 9

Simulating B&D process                       

Lab 0 (review)

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.
Competing PPs. Nearest-star distance. M/G/infinity queue. Cluster process. PP of random duration.
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.


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
Non-homogeneous Poisson process. M/G/infinity queue with uniform service time
M/G/infinity queue (continuation). Cluster process. PP of random duration
Simulating general B&D process. Linear Growth, time till absoption
LGWI, M/M/Infinity queue, N welders
Stationary distribution; absorption issues for B&D processes
Evaluation of exp(A); constituent matrices. TCMC with absorbing state(s)
Brownian motion and related formulas


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                  Solution
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  

NB:  The above solutions show only the corresponding Maple computation. This is NOT how your answers should be presented; a hand written (or better yet, typed and printed) report is required, with the corresponding Maple workspace e-mailed to your TA).


MATH 5P84   Time Series

Course Outline



Normal Distribution

Circular model

Spectral analysis

Estimating spectrum density 

Contour Integration

ARMA Trinity

Normal Approximation


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

ASSIGNMENTS   make sure to click 'reload' to get the latest version

Assgn. 0    Due:
Assgn. 1    Due:
Assgn. 2    Due:  
Assgn. 3    Due
Assgn. 4    Due:     
Assgn. 5    Due:
Assgn. 6    Due:  
Assgn. 7    Due:             
Assgn. 8    Due:                       
Assgn. 9    Due:                 


Data:     Q3.m                  Q4.m                                                                   

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

2007 FINAL


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

2014 Final        Corresponding Data          2016 Final          Corresponding Data        Maple solution

2018 Final       Corresponding Data          Maple solution             2019 Final        Corresponding Data       Maple solution            

MATH 5P88   Advanced Statistics


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


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)

ASSIGNMENTS             make sure to click 'reload' to get the latest version

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

2007 FINAL        Solution

2008 FINAL        2009 FINAL      Brief solution      2010 FINAL

2011 FINAL      Brief Solution

2012 FINAL

2013 FINAL      Maple solution         2014 FINAL

2015 FINAL      Maple solution

2016 FINAL      Maple solution

2017 FINAL      Maple Solution       2019 FINAL      Maple Solution    

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



Celestial Mechanics


The following is a Mathematica program to accompany an article of the same name, published in Vol. 20, #4 of Journal of Computational Statistics.

Population Moments of Sampling Distributions     (requires Mathematica)

PDF version of the previous program                     (for PC's without Mathematica)

Similarly, a Mathematica program to support a paper published in Vol. 34, #3 of Communications in Statistics (Simulation and Computation):

Moments of AR(1)-Model Estimators and its PDF listing 

Moments of AR(k) parameter estimators