Stats 241.3

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Stats 241.3. Probability Theory . 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:.

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### Stats 241.3

Probability Theory

Instructor:

W.H.Laverty

Office:

235 McLean Hall

Phone:

966-6096

Lectures:

T and Th 11:30am - 12:50amGEOL 255Lab: W 3:30 - 4:20 GEOL 155

Evaluation:

Assignments, Labs, Term tests - 40%Final Examination - 60%

Text:

Devore and Berk, ModernMathematical Statisticswith 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
• 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?

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.

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

Rolling a die – outcomes

• S ={ , , , , , }
• Examples
• Tossing a coin – outcomes S ={Head, Tail}

={1, 2, 3, 4, 5, 6}

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.

S

E

Rolling a die – outcomes

• S ={ , , , , , }

={ , , }

Examples

={1, 2, 3, 4, 5, 6}

E = the event that an even number is rolled

= {2, 4, 6}

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

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 ore belongs to B}

AB

A

B

Let A and B be two events, then the intersection of A and B is the event (denoted by AB ) defined by:

Intersection

A  B = {e| e belongs to A ande belongs to B}

AB

A

B

Complement

= {e| e does notbelongs to A}

A

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 (in words)

The event A or B does not occur if

the event A does not occur

and

the eventB does not occur

The event A and B does not occur if

the event A does not occur

or

the eventB does not occur

=

Another useful rule

=

In words

The event A occurs if

A occursand B occurs

or

A occurs and B doesn’t occur.

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.

We will use the notation

to mean that A is a subsetB. (B is a superset of A.)

B

A

### Union and Intersection

more than two events

Union:

E2

E3

E1

### 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 measureP 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.

R

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

S = set of all points in the target

= {(x,y)| x2 + y2≤ R2}

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

E

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).