Stats 241.3 Probability Theory

T and Th 11:30am - 12:50am GEOL 255 Lab: W 3:30 - 4:20 GEOL 155 Instructor: W.H.Laverty Office: 235 McLean Hall Phone: 966-6096 Lectures: T and Th 11:30am - 12:50am GEOL 255 Lab: W 3:30 - 4:20 GEOL 155 Evaluation: Assignments, Labs, Term tests - 40% Final Examination - 60%

Text: Devore and Berk, Modern Mathematical Statistics with applications. I will provide lecture notes (power point slides). I will provide tables. The assignments will not come from the textbook. This means that the purchasing of the text is optional.

Course Outline

Introduction Chapter 1

Probability Counting techniques Rules of probability Conditional probability and independence Multiplicative Rule Bayes Rule, Simpson’s paradox Chapter 2

Random variables Discrete random variables - their distributions Continuous random variables - their distributions Expectation Rules of expectation Moments – variance, standard deviation, skewness, kurtosis Moment generating functions Chapters 3 and 4

Multivariate probability distributions Discrete and continuous bivariate distributions Marginal distributions, Conditional distributions Expectation for multivariate distributions Regression and Correlation Chapter 5

Functions of random variables Distribution function method, moment generating function method, transformation method Law of large numbers, Central Limit theorem Chapter 5, 7

Introduction to Probability Theory Probability – Models for random phenomena

Phenomena Non-deterministic Deterministic

Deterministic Phenomena There exists a mathematical model that allows “perfect” prediction the phenomena’s outcome. Many examples exist in Physics, Chemistry (the exact sciences). Non-deterministic Phenomena No mathematical model exists that allows “perfect” prediction the phenomena’s outcome.

Non-deterministic Phenomena may be divided into two groups. Random phenomena Unable to predict the outcomes, but in the long-run, the outcomes exhibit statistical regularity. Haphazard phenomena unpredictable outcomes, but no long-run, exhibition of statistical regularity in the outcomes.

Phenomena Non-deterministic Deterministic Haphazard Random

Haphazard phenomena unpredictable outcomes, but no long-run, exhibition of statistical regularity in the outcomes. Do such phenomena exist? Will any non-deterministic phenomena exhibit long-run statistical regularity eventually?

Tossing a coin – outcomes S ={Head, Tail} Random phenomena Unable to predict the outcomes, but in the long-run, the outcomes exhibit statistical regularity. Examples Tossing a coin – outcomes S ={Head, Tail} Unable to predict on each toss whether is Head or Tail. In the long run can predict that 50% of the time heads will occur and 50% of the time tails will occur

Rolling a die – outcomes S ={ , , , , , } Unable to predict outcome but in the long run can one can determine that each outcome will occur 1/6 of the time. Use symmetry. Each side is the same. One side should not occur more frequently than another side in the long run. If the die is not balanced this may not be true.

Rolling a two balanced dice – 36 outcomes

Buffoon’s Needle problem A needle of length l, is tossed and allowed to land on a plane that is ruled with horizontal lines a distance, d, apart A typical outcome d l

Stock market performance A stock currently has a price of $125.50. We will observe the price for the next 100 days typical outcomes

Definitions

The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes. The sample space S may contain A finite number of outcomes. A countably infinite number of outcomes, or An uncountably infinite number of outcomes.

A countably infinite number of outcomes means that the outcomes are in a one-one correspondence with the positive integers {1, 2, 3, 4, 5, …} This means that the outcomes can be labeled with the positive integers. S = {O1, O2, O3, O4, O5, …}

A uncountably infinite number of outcomes means that the outcomes are can not be put in a one-one correspondence with the positive integers. Example: A spinner on a circular disc is spun and points at a value x on a circular disc whose circumference is 1. 0.0 0.1 0.9 S = {x | 0 ≤ x <1} = [0,1) x 0.2 0.8 S 1.0 0.0 [ ) 0.3 0.7 0.4 0.6 0.5

Examples Tossing a coin – outcomes S ={Head, Tail} Rolling a die – outcomes S ={ , , , , , } ={1, 2, 3, 4, 5, 6}

Rolling a two balanced dice – 36 outcomes

S ={ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} outcome (x, y), x = value showing on die 1 y = value showing on die 2

Buffoon’s Needle problem A needle of length l, is tossed and allowed to land on a plane that is ruled with horizontal lines a distance, d, apart A typical outcome d l

An outcome can be identified by determining the coordinates (x,y) of the centre of the needle and q, the angle the needle makes with the parallel ruled lines. (x,y) q S = {(x, y, q)| - < x < , - < y < , 0 ≤ q ≤ p }

An Event , E The event, E, is any subset of the sample space, S. i.e. any set of outcomes (not necessarily all outcomes) of the random phenomena S E

The event, E, is said to have occurred if after the outcome has been observed the outcome lies in E.

Examples Rolling a die – outcomes S ={ , , , , , } ={1, 2, 3, 4, 5, 6} E = the event that an even number is rolled = {2, 4, 6} ={ , , }

Rolling a two balanced dice – 36 outcomes

S ={ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} outcome (x, y), x = value showing on die 1 y = value showing on die 2

E = the event that a “7” is rolled ={ (6, 1), (5, 2), (4, 3), (3, 4), (3, 5), (1, 6)}

Special Events f = { } = the event that contains no outcomes The Null Event, The empty event - f f = { } = the event that contains no outcomes The Entire Event, The Sample Space - S S = the event that contains all outcomes The empty event, f , never occurs. The entire event, S, always occurs.

Set operations on Events Union Let A and B be two events, then the union of A and B is the event (denoted by ) defined by: AB = {e| e belongs to A or e belongs to B} AB A B

The event AB occurs if the event A occurs or the event and B occurs .

Intersection Let A and B be two events, then the intersection of A and B is the event (denoted by AB ) defined by: A  B = {e| e belongs to A and e belongs to B} AB A B

The event AB occurs if the event A occurs and the event and B occurs .

Complement Let A be any event, then the complement of A (denoted by ) defined by: = {e| e does not belongs to A} A

The event occurs if the event A does not occur

In problems you will recognize that you are working with: Union if you see the word or, Intersection if you see the word and, Complement if you see the word not.

DeMoivre’s laws  =  =

DeMoivre’s laws (in words) The event A or B does not occur if the event A does not occur and the event B does not occur The event A and B does not occur if the event A does not occur or the event B does not occur =

Another useful rule In words The event A occurs if  = In words The event A occurs if A occurs and B occurs or A occurs and B doesn’t occur.

Rules involving the empty set, f, and the entire event, S.

Definition: mutually exclusive Two events A and B are called mutually exclusive if: B A

If two events A and B are are mutually exclusive then: They have no outcomes in common. They can’t occur at the same time. The outcome of the random experiment can not belong to both A and B. B A

Some other set notation We will use the notation to mean that e is an element of A. We will use the notation to mean that e is not an element of A.

Thus

We will use the notation to mean that A is a subset B. (B is a superset of A.) B A

Union and Intersection more than two events

Union: E2 E3 E1

Intersection: E2 E3 E1

DeMorgan’s laws =

Probability

Suppose we are observing a random phenomena Let S denote the sample space for the phenomena, the set of all possible outcomes. An event E is a subset of S. A probability measure P is defined on S by defining for each event E, P[E] with the following properties P[E] ≥ 0, for each E. P[S] = 1.

P[E1] P[E2] P[E3] P[E4] P[E5] P[E6] …

Example: Finite uniform probability space In many examples the sample space S = {o1, o2, o3, … oN} has a finite number, N, of oucomes. Also each of the outcomes is equally likely (because of symmetry). Then P[{oi}] = 1/N and for any event E

Note: with this definition of P[E], i.e.

Thus this definition of P[E], i.e. satisfies the axioms of a probability measure P[E] ≥ 0, for each E. P[S] = 1.

Another Example: We are shooting at an archery target with radius R. The bullseye has radius R/4. There are three other rings with width R/4. We shoot at the target until it is hit R S = set of all points in the target = {(x,y)| x2 + y2 ≤ R2} E, any event is a sub region (subset) of S.

E, any event is a sub region (subset) of S.

Thus this definition of P[E], i.e. satisfies the axioms of a probability measure P[E] ≥ 0, for each E. P[S] = 1.

Finite uniform probability space Many examples fall into this category Finite number of outcomes All outcomes are equally likely To handle problems in case we have to be able to count. Count n(E) and n(S).