Course Code:                 MATH 2274

Course Title:                  Probability Theory I

Course Type:                  Core

Level:                              2

Semester:                        1

No. of Credits:                  3

Pre-requisite(s):               MATH1142 and MATH 1151.

Course Rationale

Probability Theory is now an indispensable tool in many applications of Mathematics to the Natural Sciences, Computing, Finance, Insurance and, of course, to Statistics. It is hard to imagine someone graduating with a degree in Mathematics that does not include a course in Probability Theory. This course provides Mathematics Majors with the basics of Probability Theory (both discrete and continuous).

It should be noted that the Ministry of Education insists that teachers include such a course in their degree, otherwise they are not paid as graduate teachers.

Course Description

This is an introductory course that approaches probability theory from two perspectives:

• Probability theory is a branch of mathematics. As such, we will focus on the fundamental assumptions of Probability Theory and how the main properties of Probability Measures proceed from these assumptions. Throughout the course, therefore, students will be expected to be able to derive the main results that they use. Very little will be assumed without proof.
• Probability Theory is primarily concerned with modelling phenomena with uncertain outcomes.  The course emphasizes this. It is most definitely not a course in Pure Mathematics.

A knowledge of calculus (including a good understanding of limits, continuity, differentiability) is assumed (hence the need for Math1150). An appreciation of the idea of proof is expected but Math1140 is not essential (though it is desirable).

The course begins with a discussion of the basic ideas of probability, including the axioms of probability, combinatorial probability, conditional probability and independence. The rest of the course focuses on distribution theory. The distribution theory of one discrete and one continuous random variable is discussed. Special attention is paid to well-known discrete and continuous distributions such as the Bernoulli, Binomial, Poisson, Exponential, Gamma and Normal. Then the distribution theory of several random variables is discussed.  The idea of a statistic is introduced and the distribution theory of the mean and the sample variance is described. This leads finally to the idea of convergence in distribution and the Central Limit Theorem (without proof)

The approach taken is non-rigorous. In particular, there will be no mention of sigma algebras or of measure theory.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly assignments and problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the weekly assignments and in-course tests followed by a final examination based on the whole course

Learning Outcomes

Upon successful completion of the course, students will be able to:

• Derive basic properties of probability measures from the axioms of probability
• Use simple combinatorial methods to compute the probability of events in experiments with a finite number of outcomes.
• Use the ideas of conditional probability and independence and Bayes Theorem to compute the probability of an event.
• Given the probability function or probability density function of one random variable, compute the distribution function and moments of the random variable (including the Bernoulli, Binomial, Poisson, Gamma and Normal random variables).
• Given the joint pf/pdf of several random variables, compute the probabilities of events, expectations and the covariances and correlations between any two random variables.
• Use the properties of linear combinations or sums of independent Bernoulli, Poisson, Gamma and Normal random variables.
• Derive the main properties of the sample mean and sample variance, with special attention to the normal and Bernoulli distributions.
• Define the notions of convergence in distribution.
• State the Central Limit Theorem (CLT) and its immediate corollaries.
• Use the CLT in simple situations involving sums and means of random variables.

Content

• Basic Ideas of Probability: Definition of statistical experiment, sample space, events; The Calculus of Events; equally likely events; combinatorial probability; definition of conditional probability; application to computing probabilities in simple situations; the Theorem of Total Probability and Bayes Theorem; independent events; applications to simple situations including systems of components in series and in parallel (8 hours).
• Discrete Random Variables: Definition of a random variable; definition and examples of discrete and continuous random variables; the probability function and distribution function of a discrete random variable; definition and calculation of the expectation, variance and moments of a discrete random variable from the probability function; detailed properties of the Bernoulli, Binomial, Hypergeometric, Geometric and Poisson random variables; the Poisson approximation to the Binomial (6 hours).
• Continuous random variables: Probability density function and distribution function of one continuous random variable; calculating the probability of an event from the pdf; percentiles of a continuous r.v.; expectation and moments of a continuous random variable; the pdf and moments of the exponential, normal, gamma and chi-squared random variables; properties of one normal random variable; the normal approximation to the binomial; the distribution of X given X>a; the memoryless property of the exponential distribution; the Poisson process; the distribution of functions of one discrete or continuous random variable; the distribution function of any random variable (10  hours).
• Several Random Variables The joint distribution of several random variables in the discrete and continuous case; joint pdf; evaluating probabilities of events using the joint pdf of two random variables; marginal and conditional distributions; independence of random variables; expectation and its properties; E(XY)=E(X)E(Y) when X and Y are independent; covariance and correlation; the mean and variance of linear combinations of several random variables; the distribution of linear combinations of independent normal random variables and simple applications. (6 hours).
• Sample Statistics Definition of a statistic; the definition and distribution of the sample mean and the sample variance; special case when the population is normal; the Central Limit Theorem and its applications to simple problems. (4 hours).

Teaching Methodology

Lectures, tutorials, assignments and problem papers.

Lectures: Three (3) lectures each week (50 minutes each). Tutorials as required.

Assignments: One assignment (marked) per week.

Additional problems will be given during lectures and tutorials but will not be marked. However, students will need to do some of these, as well as the assignments, in order to learn the material properly and to adequately prepare for examinations and quizzes.

Assessment

Coursework: 50%, based on three in-term examinations (40) and assignments (10%).

Final Examination: 50% (one 2-hour written paper).

Course Calendar

WEEK

TOPIC

ASSIGNMENT

ASSESSMENT

1

Introduction/Course Overview

Basic Ideas of Probability: Axioms of Probability; Combinatorial probability

Assignment 1 handed out

None

2

More Combinatorial Probability; Conditional Probability, Bayes Theorem  and the Theorem of Total Probability

Assignment 1 returned.

Assignment 2 handed out

Quiz 1

(On Week 1)

3

Discrete Random Variables: Definition of a random variable, Probability Function, expectation, variance, Bernoulli, Binomial random variables.

Assignment  3

Assignment 2 returned.

Quiz 2

(on Week 2)

4

Discrete Random Variables: The Poisson random variable, Poisson approximation to the Binomial

Assignment 4  Assignment 3 returned

Test 1 on Topics in

Weeks 1 and 2.

5

Continuous Random Variables: probability density function and density function  of a continuous random variables, expectation and moments, normal distribution, normal approx to the Binomial

Assignment  5

Assignment  4 returned

Quiz 3

(Discrete Random Variables)

6

Continuous Random Variables: Gamma. Exponential and chi-squared distributions, the Poisson process, transformations of one random variable.

Assignment 6

Assignment 5 returned

Quiz 4

(Week 5)

7

Continuous random variables: density function of any random variable;

Assignment 7

Assignment 6 Returned

Quiz 5

(Week 6)

8

Several Random variables: joint probability function and probability density function, evaluation of probabilities using joint probability density function.

Assignment  8

Assignment 7 returned

Quiz 6

(Continuous Random Variables, Week 7)

9

Independence; expectation; covariance and correlation

Assignment  9

Test 2

(Discrete and Continuous Random Variables)

10

Several Random Variables: covariance and correlation; linear combinations of several random variables

Assignment 10

Assignment 9 returned

Quiz 7

(Weeks 8,9)

11

Sample Statistics: the sample mean and sample variance; special case when population is normal;

Assignment  11

Quiz 8

(Week 10)

12

The Central Limit Theorem and its applications

Assignment  12

Test3:(Several Random Variables, Sample Statistics)

13

Revision

Revision

None

Essential Texts:

An Introduction to Probability Theory, DMCS, Robin Antoine, 2003.

Other Recommended Texts:

A First Course in Probability Theory, Prentice- Hall, Ross, Sheldon, 2009.

Probability and Statistical Inference (8th Edition) Prentice-Hall, R. V. Hogg & E. Tanis, 2009.