Stats 241.3

Probability Theory

Instructor: W.H.Laverty

Office: 235 McLean Hall

Phone: 966-6096

Lectures: M T W Th F 1:30pm - 2:50pmThorv 105

Lab: T W Th 3:00 - 3:50 Thorv 105

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

Test: Every Thursday in lab

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• Chapters 5, 6

Introduction to Probability Theory

Probability – Models for random phenomena

Phenomena

Deterministic Non-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.

1. Random phenomena– Unable to predict the outcomes, but in the long-

run, the outcomes exhibit statistical regularity.

2. Haphazard phenomena– unpredictable outcomes, but no long-run,

exhibition of statistical regularity in the outcomes.

Phenomena

Deterministic

Non-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.

Examples1. 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

2. 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.

3. Rolling a two balanced dice – 36 outcomes

4. 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

d

l

A typical outcome

0

50

100

150

200

250

0 20 40 60 80 100

day

pric

e

5. 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

1. A finite number of outcomes.2. A countably infinite number of outcomes, or3. 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.

0.00.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9x S = {x | 0 ≤ x <1} = [0,1)

0.0 1.0[ )

S

Examples1. Tossing a coin – outcomes S ={Head, Tail}

2. Rolling a die – outcomes S ={ , , , , , }

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

3. 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 1y = value showing on die 2

4. 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

d

l

A typical outcome

An outcome can be identified by determining the coordinates (x,y) of the centre of the needle and θ, the angle the needle makes with the parallel ruled lines.

(x,y) θ

S = {(x, y, θ)| -∞ < x < ∞, - ∞ < y < ∞, 0 ≤ θ ≤ π }

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

SE

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

SE

Examples

1. Rolling a die – outcomes S ={ , , , , , }

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

E = the event that an even number is rolled= {2, 4, 6}

={ , , }

2. 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 1y = 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 - φ

φ = { } = the event that contains no outcomes

The Entire Event, The Sample Space - S

S = the event that contains all outcomes

The empty event, φ , never occurs.The entire event, S, always occurs.

Set operations on EventsUnionLet 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

BA∪

BA∪

BA∪

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

A(B

A B

BA∪

BA∪

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

BA∩

BA∩

BA∩

A'B

A B

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

BA∩

BA∩

Complement

Let A be any event, then the complement of A (denoted by ) defined by:

= {e| e does not belongs to A}A

A

AA

The event occurs if the event A does not occur

A

AA

In problems you will recognize that you are working with:

1. Union if you see the word or,2. Intersection if you see the word and,3. Complement if you see the word not.

1. A B A B∪ = ∩

2. A B A B∩ = ∪

DeMorgan’s laws

=

=

∩

∪

1. A B A B∪ = ∩

2. A B A B∩ = ∪

DeMorgan’s laws (in words)

The event A or B does not occur if the event A does not occurandthe event B does not occur

=

The event A and B does not occur if the event A does not occurorthe event B does not occur

∪

Another useful rule

=

In wordsThe event A occurs if A occurs and B occursorA occurs and B doesn’t occur.

( ) ( ) A A B A B= ∩ ∪ ∩

1. A Aφ∪ =

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

2. A φ φ∩ = 3. A S S∪ =

4. A S A∩ =

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

A B φ∩ =

A B

If two events A and B are are mutually exclusive then:

A B

1. They have no outcomes in common.They can’t occur at the same time. The outcome of the random experiment can not belong to bothA and B.

Some other set notation

We will use the notatione A∈

to mean that e is an element of A.

We will use the notatione A∉

to mean that e is not an element of A.

Thus

{ } or A B e e A e B∪ = ∈ ∈

{ } and A B e e A e B∩ = ∈ ∈

{ }A e e A= ∉

We will use the notation

(or )A B B A⊂ ⊃

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

. . if then .e A e B∈ ∈i e

BA

Union and Intersection

more than two events

Union: 1 2 31

k

i ki

E E E E E=

= ∪ ∪ ∪ ∪∪

E1

E2

E3

1 2 31

ii

E E E E∞

=

= ∪ ∪ ∪∪

Intersection: 1 2 31

k

i ki

E E E E E=

= ∩ ∩ ∩ ∩∩

E1

E2

E3

1 2 31

ii

E E E E∞

=

= ∩ ∩ ∩∩

1. i ii i

E E⎛ ⎞=⎜ ⎟

⎝ ⎠∪ ∩

DeMorgan’s laws

=

2. i ii i

E E⎛ ⎞=⎜ ⎟

⎝ ⎠∩ ∪

Probability

Suppose we are observing a random phenomenaLet 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

1. P[E] ≥ 0, for each E.2. P[S] = 1.

3. [ ] if for all ,i i i iii

P E P E E E i jφ⎡ ⎤= ∩ =⎢ ⎥

⎣ ⎦∑∪

[ ] [ ] [ ]1 2 1 2P E E P E P E∪ ∪ = + +

[ ]i iii

P E P E⎡ ⎤=⎢ ⎥

⎣ ⎦∑∪

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

[ ] ( )( )

( ) no. of outcomes in =total no. of outcomes

n E n E EP En S N

= =

( ): = no. of elements of n A ANote

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

[ ] ( )( )

( ) no. of outcomes in =total no. of outcomes

n E n E EP En S N

= =

[ ]1. 0.P E ≥

[ ] ( )( )

2. = 1n S

P Sn S

=

( ) ( )1 23. i

ii

i

n En E n E

P EN N

⎛ ⎞⎜ ⎟ + +⎡ ⎤ ⎝ ⎠= =⎢ ⎥

⎣ ⎦

∪∪

[ ] [ ]1 2 if i jP E P E E E φ= + + ∩ =

Thus this definition of P[E], i.e.

[ ] ( )( )

( ) no. of outcomes in =total no. of outcomes

n E n E EP En S N

= =

satisfies the axioms of a probability measure

1. P[E] ≥ 0, for each E.2. P[S] = 1.

3. [ ] if for all ,i i i iii

P E P E E E i jφ⎡ ⎤= ∩ =⎢ ⎥

⎣ ⎦∑∪

[ ] [ ] [ ]1 2 1 2P E E P E P E∪ ∪ = + +

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.

E

[ ] ( )( )

( )2: =

Area E Area EP E

Area S Rπ=Define

[ ]

2

2

1416

R

P BullseyeR

π

π

⎛ ⎞⎜ ⎟⎝ ⎠= =

[ ]

2 2

2

3 29 4 54 4 16 16

R RP White ring

R

π π

π

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟ −⎝ ⎠ ⎝ ⎠= = =

Thus this definition of P[E], i.e.

satisfies the axioms of a probability measure

1. P[E] ≥ 0, for each E.2. P[S] = 1.

3. [ ] if for all ,i i i iii

P E P E E E i jφ⎡ ⎤= ∩ =⎢ ⎥

⎣ ⎦∑∪

[ ] [ ] [ ]1 2 1 2P E E P E P E∪ ∪ = + +

[ ] ( )( )

( )2=

Area E Area EP E

Area S Rπ=

Finite uniform probability space

Many examples fall into this category1. Finite number of outcomes 2. All outcomes are equally likely

3. [ ] ( )( )

( ) no. of outcomes in =total no. of outcomes

n E n E EP En S N

= =

( ): = no. of elements of n A ANote

To handle problems in case we have to be able to count. Count n(E) and n(S).

Next Topic: Counting