01+S241+Introduction to+Probability

8/3/2019 01+S241+Introduction to+Probability

1/67

Stats 241.3

Probability Theory


2/67

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%


3/67

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.


4/67

Course Outline


5/67

Introduction

Chapter 1


6/67

Probability

Counting techniques

Rules of probability

Conditional probability and independenceMultiplicative Rule

Bayes Rule, Simpsons paradox

Chapter 2


7/67

Random variables

Discrete random variables - their distributions

Continuous random variables - theirdistributions

ExpectationRules of expectation

Momentsvariance, standard deviation, skewness,

kurtosisMoment generating functions

Chapters 3 and 4


8/67

Multivariate probability distributions

Discrete and continuous bivariate distributions

Marginal distributions, Conditional

distributions

Expectation for multivariate distributions

Regression and Correlation

Chapter 5


9/67

Functions of random variables

Distribution function method, moment

generating function method, transformation

method

Law of large numbers, Central Limit theorem

Chapter 5, 7


10/67

Introduction to Probability

Theory

ProbabilityModels for random

phenomena


11/67

Phenomena

DeterministicNon-deterministic


12/67

Deterministic Phenomena

There exists a mathematical model that allows

perfect prediction the phenomenasoutcome.

Many examples exist in Physics, Chemistry

(the exact sciences).Non-deterministic Phenomena

No mathematical model exists that allows

perfect prediction the phenomenasoutcome.


13/67

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 theoutcomes.


14/67

Phenomena

Deterministic

Non-deterministic

Haphazard

Random


15/67

Haphazard phenomena

unpredictable outcomes, but no long-run,

exhibition of statistical regularity in theoutcomes.

Do such phenomena exist?

Will any non-deterministic phenomena exhibitlong-run statistical regularity eventually?


16/67

Random phenomena

Unable to predict the outcomes, but in the long-

run, the outcomes exhibit statistical regularity.

Examples

1. Tossing a coinoutcomes S ={Head, Tail}Unable to predict on each toss whether is Head or

Tail.

In the long run can predict that 50% of the timeheads will occur and 50% of the time tails will occur


17/67

2. Rolling a dieoutcomes

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.


18/67

3. Rolling a two balanced dice36 outcomes


19/67

4. Buffoons Needle problem

A needle of length l, is tossed and allowed to land

on a plane that is ruled with horizontal lines adistance, d, apart

d

l

A typical outcome


20/67

0

50

100

150

200

250

0 20 40 60 80 100

day

price

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


21/67

Definitions


22/67

The sample Space, S

The sample space, S, for a random phenomenais 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, or

3. An uncountably infinite number of outcomes.


23/67

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 labeledwith the positive integers.

S = {O1, O2, O3, O4, O5, }


24/67

A uncountably infinite number of outcomes

means that the outcomes are can not be put in a

one-one correspondence with the positiveintegers.

Example: A spinner on a circular disc is spun

and points at a valuex on a circular disc whosecircumference is 1.

0.00.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9x S = {x | 0 x


25/67

Examples

1. Tossing a coinoutcomes S ={Head, Tail}


S ={ , , , , , }

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


26/67



27/67

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


28/67

4. Buffoons Needle problem

A needle of length l, is tossed and allowed to land

on a plane that is ruled with horizontal lines adistance, d, apart

d

l

A typical outcome


29/67

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 parallelruled lines.

(x,y) q

S = {(x,y, q)| -


30/67

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


31/67

The event,E, is said to have occurred if after

the outcome has been observed the outcome lies

inE.

S

E


32/67

Examples


S ={ , , , , , }

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

E= the event that an even number is

rolled

= {2, 4, 6}

={ , , }


33/67



34/67

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


35/67

E= the event that a 7 is rolled

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


36/67

Special Events

The Null Event, The empty event - ff= { } = 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.


37/67

Set operations on Events

UnionLetA andB be two events, then the union ofA

andB is the event (denoted by ) defined by:

AB = {e| e belongs toA ore belongs toB}

AB

A B

BA

BA

BA


38/67

The eventAB occurs if the eventA occurs or

the event andB occurs .

A

B

A B

BA

BA


39/67

IntersectionLetA andB be two events, then the intersection

ofA andB is the event (denoted byAB ) defined

by:

AB = {e| e belongs toA ande belongs toB}

AB

A B

BA

BA

BA


40/67

AB

A B

The eventAB occurs if the eventA occurs and

the event andB occurs .

BA

BA


41/67

ComplementLetA be any event, then the complement ofA

(denoted by ) defined by:

= {e| e does notbelongs toA}A

A

A

A


42/67

The event occurs if the eventA does not

occur

A

A

A


43/67

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.


44/67

1. A B A B

2. A B A B

DeMoivres laws

=

=


45/67

1. A B A B

2. A B A B

DeMoivres laws (in words)

The eventA orB does not occur if

the eventA does not occur

and

the eventB does not occur

=

The eventA andB does not occur ifthe eventA does not occur

or

the eventB does not occur


46/67

A A B A B

Another useful rule

=

In words

The eventA occurs ifA occursandB occursor

A occurs andB doesnt occur.

Rules involving the empty set f and


47/67

1. A Af

Rules involving the empty set, f, and

the entire event, S.

2. A f f

3. A S S

4. A S A


48/67

Definition: mutually exclusive

Two eventsA andB are called mutuallyexclusive if:

A B f

A B

If A d B ll


49/67

If two eventsA andB are are mutually

exclusive then:

A B

1. They have no outcomes in common.They cant occur at the same time. The outcome

of the random experiment can not belong to both

A andB.


50/67

Some other set notation

We will use the notatione A

to mean that e is an element ofA.

We will use the notatione A

to mean that e is not an element ofA.


51/67

Thus orA B e e A e B

andA B e e A e B

A e e A


52/67

We will use the notation(or ) A B B A

to mean thatA is a subsetB. (B is a superset ofA.)

. . if then .e A e B

i e

B

A


53/67

Union and Intersection

more than two events

k


54/67

Union: 1 2 31

i ki

E E E E E

E1

E2

E3

1 2 3

1

ii

E E E E

k


55/67

Intersection: 1 2 31

i ki

E E E E E

E1

E2

E3

1 2 3

1

ii

E E E E

D M l


56/67

1.i i

i i

E E

DeMorgans laws

=

2.i i

i i

E E


57/67

Probability

Suppose we are observing a random phenomena


58/67

Suppose we are observing a random phenomena

Let S denote the sample space for the phenomena, the

set of all possible outcomes.

An eventEis a subset ofS.

A probability measureP is defined on S by defining

for each eventE, P[E] with the following properties

1. P[E] 0, for eachE.

2. P[S] = 1.

3. if for all ,i i i iii

P E P E E E i jf

1 2 1 2P E E P E P E


59/67

i iii

P E P E

P[E1] P[E2]P[E3]

P[E4]

P[E5

]P[E6]

Example: Finite niform probabilit


60/67

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/Nand for any eventE

no. of outcomes in=total no. of outcomes

n E n E EP E

n S N

: = no. of elements ofn A ANote

N t i h hi d fi i i f P[E] i


61/67

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

no. of outcomes in= total no. of outcomes

n E n E E

P E n S N

1. 0.P E

2. = 1n S

P Sn S

1 23.i

ii

i

n E

n E n E P EN N

1 2 if i jP E P E E E f

Thus this definition of P[E] i e


62/67

Thus this definition ofP[E], i.e.

no. of outcomes in= total no. of outcomes

n E n E E

P E n S N

satisfies the axioms of a probability measure


2. P[S] = 1.


P E P E E E i jf

1 2 1 2P E E P E P E

Another Example:


63/67

Another Example:

We are shooting at an archery target with radius

R. The bullseye has radiusR/4. There are threeother rings with widthR/4. We shoot at the target

until it is hit

R

S = set of all points in the target

= {(x,y)|x2 +y2R2}

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


64/67

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

E

2

: = Area E Area E

P E Area S Rp

Define


65/67

2

2

14

16

R

P Bullseye R

p

p

2 2

2

3 2

9 4 54 4

16 16

R R

P White ring

R

p p

p

Thus this definition of P[E] i e


66/67

Thus this definition ofP[E], i.e.

satisfies the axioms of a probability measure


2. P[S] = 1.


P E P E E E i jf

1 2 1 2P E E P E P E

2=

Area E Area E

P E Area S Rp


67/67

Finite uniform probability space

Many examples fall into this category

1. Finite number of outcomes

2. All outcomes are equally likely

3.

no. of outcomes in=

total no. of outcomes

n E n E EP E

n S N

: = no. of elements ofn A ANote

To handle problems in case we have to be able to

Documents

01+S241+Introduction to+Probability