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MATH 1P98 Basic Statistical Methods
Review for Test 2 (2004)
MATH 2F05 Applied Advanced Calculus
CLASS TRANSPARENCIES: Case of multiple eigenvalues (suppliment)
FIRST MIDTERM TEST:
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.
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!
LAB 0: Basics of Maple LAB 1: Exploring, graphically, families
of solutions to 1st-order ODE
LAB 2: Verifying solutions. Solving and visualizing
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
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 . y + r by Undetermined Coefficients and Variation
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 19 Complex functions - differentiation and
ASSIGNMENTS AND SOLUTIONS Assgn. 1 Due: Sept. 23 Sol. 1
Due: Sept. 30
Assgn. 3 Due: Oct. 7 Sol. 3
Due Oct. 14
Due Oct. 21
Due Oct. 28
Assgn. 7 Due Nov.
4 Sol. 7
Due Nov. 11
Due Nov. 18
Due Nov. 25
Due Dec. 2
Due Jan. 6
Due Jan. 13
Due Jan. 20
Due Jan. 27
Due Feb. 3
Assgn. 17 Due Feb. 10 Sol. 17
Due Feb. 24
Due March 10
Due March 24 Sol. 21
Due April 5 Sol. 22
MATH 2P81 Probability
Two regular-size sheets (both
sides) of notes and loading of a single Maple workspace
(of any size) is allowed.
Any questions concerning marking (midterms
and assignments) should be directed to your TA who has the authority to correct
any potential mistakes. Only if agreement cannot be
reached, the student has the option of having the WHOLE
exam (not just specific questions) marked by the instructor
(the new mark will stand). It is then the student's responsibility
to ask the TA to update his records accordingly.
Each solution will be available only after
the corresponding assignment's deadline.
OLD TEST 1 (includes a solution sheet)
ANOTHER TEST 1
2007 First Midterm
2008 First Midterm
2010 First Midterm
2010 Substitute Midterm
2011 First Midterm
2012 First Midterm
OLD TEST 2 (with solution)
ANOTHER TEST 2
2007 Second Midterm
2009 Second Midterm
2010 Second Midterm
2011 Second Midterm
2012 Second Midterm
Final Exam 2006
Final Exam 2007
Final Exam 2008
Final Exam 2009
Final Exam 2010
Final Exam 2011
Final Exam 2012
Maple's treatment of special discrete
Continuous distributions by
MATH 2P82 Mathematical
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,
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
ASSIGNMENTS AND SOLUTIONS
DATA FOR: 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
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
Fitting a function by a polynomial
utilizing Chebyshev polynomials.
Trapezoidal and Simpson's (composite) rules;
Designing 3- and more-point formulas, without
and with a weight function.
Gaussian formulas, without and with a weight
Designing formulas for numerical differentiation;
Solving one or more non-linear equations.
Second-order ODE (linear and non-linear)
with given boundary values.
fit (discrete data)
fit to functions (Legendre polynomials)
fit with weights (Chebyshev polynomials)
(trapezoidal rule); Romberg's technique
own formulas. Improper integrals
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
technique for solving (a set of) ODEs
Trigonometric (Fourier) approximation;
continuous and discrete case
polynomials (Newton and Lagrange technique)
least-square discrete fit (to a set of point); weights;
LS continuous (ie. to functions) fit; Gram-Schmidt and
Improved (weighted) LS fit using
elimination with full pivoting
Gram-Schmidt; composite trapezoidal
rule; Romberg algorithm
Other formulas. Integrals with
a weight function
LU decomposition; boundary-value
ODE; Richardson exptrapolation
Non-linear, second-order ODE
Solving ODE of any order -
initial value problem
to various function
polynomial; Fast Fourier transform
ASSIGNMENTS AND SOLUTIONS Assgn.
1 Due: Sept. 23
2 Due: Oct. 3, 11am Sol. 2
3 Due: Oct. 11, 2pm Sol. 3
4 Due: Oct. 17, 11am. Sol. 4
Assgn. 5 Due: Oct. 24, 11am Sol. 5
Due Oct. 31, 11am
7. Due Nov. 8, 2pm Sol. 7
8 Due Nov. 14, 11am Sol. 8
9 Due Nov. 21, 11am
Assgn. 10 Due Nov. 28,
11am Sol. 10
Assgn. 11 Due Dec. 5, 11am
MATH 3P82 Regression Analysis
MATH 4F83 Statistics
SAMPLE PROGRESS EXAMS
SAMPLE 2nd MIDTERM EXAMS
SAMPLE FINAL EXAMS
MATH 4P84 - Topics in Stochastic Processes and Models
MATH 4P85 - Topics in Advanced Statistics
Second midterm ( open book): March 17th, in
Lab. Covers Assgn. 2 and 3, i.e. Chapter 7
ASSIGNMENTS (numerical answers should be given
to 4 significant digits - make sure to click 'reload'
to get the latest version)
Assgn. 0 ( Due: Jan 21) Sol. 0
Assgn. 1 ( Due: Feb 4) Sol. 1
Assgn. 2 ( Due: Feb 28)
Assgn. 3 ( Due: March 11)
Assgn. 4 ( Due: March 20) Sol. 4
Assgn. 5 ( Due: ) Sol. 5
2014 LABS Maple basics;
PGF; generating random samples; composite distributions.
process (two ways). Time till next arrival.
from Uniform and Poisson distributions. Non-homogeneous
PP. Competing PPs. Nearest-star distance. M/G/infinity
queue. Cluster process. PP of random duration.
Special case of Pure Death process.
process. Simulating a general B&D Markov process. Time
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.
Yule and other Autoregressive models.
OLD LABS Examples of
PDF, MGF, PGF, RIS and CLT
formula, marginal and conditional PDF, generating random
#s from various distributions Simulating Poisson
process in two different ways; time till the next arrival
Poisson process. M/G/infinity queue with uniform service
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
Another First Midterm
2005 First Midterm Solution
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)
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
Solution 2008 FINAL Maple Solution 2009 FINAL 2010 FINAL Maple Solution 2011 FINAL Maple "Solution" 2012 FINAL Maple "solution" 2013 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 report is required,
with the corresponding Maple workspace attached to it (or
e-mailed to email@example.com; don't forget to include your
full name in the subject field).
STUDIES MATH 5P84 Time Series
Multivariate Normal distribution,
partial correlation coefficient
Distribution of sample correlation
Solving (through eigenvectors)
V = A.At (don't use in your Assgn. 1)
Data file called "assgn4.m" to be used with Q1;
the stored list is called Y Assgn.
5 Due: Data file called "assgn5.m"
7 Due: Data for Questions
4 and 5 (stored as list
Assgn. 8 Due:
Assgn. 9 Due: Data for Question 4 (sequence X)
Data: Q3.m Q4.m
Maple 'solution' (to be presented
in proper manner)
DATA: Q_3.m Q_4.m
Data (always called X):
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.
Data for Question 3 (list W) Data for Question 4 (list Y) Data for Question 5 (list X) Maple solution
MATH 5P88 Advanced Statistics
Lectures moved to Tuesdays & Thursdays 9:00
- 10:30, in J409A
LECTURE NOTES Introduction
to Large-sample Theory
ML estimator Extending
Central Limit Theorem
Generating random numbers from
Linear congruential generator
Beyond Normal Approximation -
expanded into SIPs
to Gamma and Poisson distributions
Approximating the r distribution
Example of bivariate approximation
estimators - gamma distribution
estimators of parameters of gamma distribution
Carlo simulation of a sample from a distribution
All key steps and answers must
be entered, clearly and legibly, in the examination booklet.
NO MAPLE ACCEPTED IN ANY FORM!
The following is a Mathematica program to accompany
an article of the same name, published in Vol. 20,
#4 of Journal of Computational Statistics.
Similarly, a Mathematica
program to support a paper published in Vol. 34, #3 of Communications in
Statistics (Simulation and Computation):