STAT 3502 Class 1 Notes

8/9/2019 STAT 3502 Class 1 Notes

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STAT 3502

Introduction to Probability and Statistics for

Engineers

Instructor: Christopher Gravel

Carleton University

STAT 3502

C. Gravel (Carleton) STAT 3502 January 2015 1 / 26


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Introduction

Introduction

Statistics

Statistics is the science of learning from data, and of measuring,controlling, and communicating uncertainty." American StatisticalAssociation



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Chapter 2 - Probability

Chapter 2 - Probability

Probability Theory

The subject of probability theory is the foundation upon which all ofstatistics is built, providing means for modelling populations,experiments, or almost anything that can be considered a randomphenomenon." Casella and Berger



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Chapter 2 - Probability 2.1 Sample Spaces and Events

2.1 Sample Spaces and Events

Definition: Experiment (p. 51)

An experiment is any activity or process whose outcome is subjectto uncertainty.

Definition: Sample Space (p. 51)

The sample space of an experiment, denoted by S, is the set of allpossible outcomes of that experiment.



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Sample Spaces and Events - 2

Definition: Event (p. 52)

An event is any collection (subset) of outcomes contained in thesample space, S. An event is simpleif it consists of exactly one

outcome and compound if it consists of more than one outcome.

When an experiment is performed, a particular event A is said to

occurif the resulting experimental outcome is containedin A.



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Example: A deck of cards

Consider the following three events based on 1 card drawn from adeck of cards:

A: a heart is drawnB: a non-face card is drawn

C: a card is drawn whose number if divisible by 3

Identify the sample space, the simple events and the compoundevents.


h b bili l d


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Example: The Sample Space

which we will denote as:

S= {A,2,3,...,J, Q, K,A,2,3,...,J, Q, K,

A,2,3,...,J, Q, K,

A,2,3,...,J, Q, K}


Ch t 2 P b bilit 2 1 S l S d E t


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Example: The Events

The simple events are the elements ofS, or all the cards in thedeck.

The compound events are:

A = {A,2,3,...,J, Q, K}

B = {2,3,...,9,10,2,3,...,9,10,

2,3,...,9,10,2,3,...,9,10}

C = {3,6,9,3,6,9,3,6,9,3,6,9,}

See Example 2.5 p. 52 for an additional example.


Chapter 2 Probability 2 1 Sample Spaces and Events


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Since an event can be thought of as a set, we can use relationshipsand results from set theory to study events. The followingoperations will be used to create new events from given events.

Definition: Set Operations (p.53)

LetA and B be 2 events contained in S.

Complement: The complement ofA is the set of all outcomes

in S NOTcontained in A. Denoted as A

or Ac.

Union: The event of all outcomes either in A OR B. Denoted as

A B. Intersection: The event of all outcomes in A AND B. Denoted

as A B.


Chapter 2 Probability 2 1 Sample Spaces and Events


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Venn Diagram

A Venn diagram is a logical schematic that demonstrates all possiblelogical relations between finite collections of sets.

See p.54 for some discussion on Venn diagrams in the book.


Chapter 2 - Probability 2 1 Sample Spaces and Events


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Chapter 2 Probability 2.1 Sample Spaces and Events


Recall, the following compound events:A: a heart is drawn

B: a non-face card is drawn

C: a card is drawn whose number if divisible by 3

Set Operations

The list outcomes in the following events:

i) B

ii) A B

iii) A C

iv) B C




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Chapter 2 Probability 2.1 Sample Spaces and Events


Definition: Mutually Exclusive/Disjoint (p. 54)Let denote the null event (the event consisting of no outcomeswhatsoever). When A B= , A and B are said to be mutuallyexclusive ordisjoint events. is also known as the empty set.

In the previous example, we demonstrated that B C= , since

there areno outcomes in common between a face card" ANDthe card drawn is divisible by 3".

See Example 2.10 p. 54 for an additional example.




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p y p p

Additional Examples

Q. 8, P. 55An engineering construction firm is currently working on powerplants at three different sites. Let A denote the event that the plantat site is completed by the contract date. Use the operations ofunion, intersection, and complementation to describe each of the

following events in terms ofA1, A2, and A3, draw a Venn diagram,and shade the region corresponding to each one.

At least one plant is completed by the contract date.

All plants are completed by the contract date.

Only the plant at site 1 is completed by the contract date.

Exactly one plant is completed by the contract date.

Either the plant at site 1 or both of the other two plants arecompleted by the contract date.




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Additional Examples

Q. 9, P. 55 DeMorgans Laws

Use Venn diagrams to verify the following two relationships for any

events A and B (these are called De Morgans laws):

(A B) =A B

(A B) =A B


Chapter 2 - Probability 2.2 Axioms, Interpretations, and Properties of Probability


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2.2 Axioms, Interpretations, and Properties of

ProbabilityGiven an experiment and a sample space S, the objective of

probability is to assign to each event A a number P(A), called theprobability of the event A, which will give a precise measure of thechance thatA will occur.

The Axioms of Probability, p. 56

i) For any event,A, P(A) 0.ii) P(S) = 1.

iii) IfA1, A2, A3... is an infinite collection of disjoint events, then,

P(A1 A2 A3 ...) =

=1P(A

)

Notation: P(A1 A2 A3 ...) =P

=1

A

Infinite collection of disjoint events can be written as, A Aj =

for all , j such that =j.C. Gravel (Carleton) STAT 3502 January 2015 15 / 26



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Justification and Implications of the Axioms

The axioms are meant to be the starting point of reasoning" fromwhich to build probability theory. As such, they will ) not containany property that can be derived from others on the list and ) canbe justified by intuition.

Justification

i) The chance ofA occurring is non-negative.

ii) The probability of ALL outcomes occurring is 1 (100%).

iii) Assume no 2 events can occur simultaneously (disjointness).Then, the probability of at least 1 event occurring is the sum ofthe probabilities of the individual events.




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Justification and Implications of Axioms - 2i) P() = 0.

ii) Axiom 3 holds for a finite collection of disjoint sets:

P

k=1

A

=

k=1

P(A).

iii) For any eventB, P(B) = 1 P(B) which implies P(B) + P(B

) = 1.

iv) For any event A, 0 P(A) 1.v) For any 2 events (not necessarily disjoint),A and B,

P(A B) =P(A) + P(B) P(A B).

Note, ifA and B are disjoint, P(A B) = 0, andP(A B) =P(A) + P(B), see axiom 3.

vi) For any 3 events A, B, and C,

P(A B C) = P(A) + P(B) + P(C) P(A B) P(B C)

P(A C) + P(A B C)

See page 61. for Venn diagram of (vi). Try to convince yourself

of the formula.C. Gravel (Carleton) STAT 3502 January 2015 17 / 26



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Interpreting and Computing Probabilities - 1

Equiprobable EventsIf each possible event, E for every = 1,...,n, is equally likely(equiprobable) then the probability of each event is,

p= P(E) =1

n.

Example: A deck of cards.

i) The probability of a 3 of hearts is 1/52.

ii) The probability of a queen of spades is 1/52.

and so on...



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i) Let A be the event of drawing a heart. Then,

P(A) =N(A)

n=

13

52= 0.25.

ii) LetB be the event of drawing a non face card. Then,

P(B) =N(B)

n=

36

52= 0.6923.

iii) Let C be the event of drawing a card displaying a numberdivisible by 3. Then,

P(C) =N(B)

n=

12

52= 0.2308.

and so on... Try determining the probabilities of the events listedin the Sample Spaces and Events - 8 slide:

i) B

ii) A Biii) A Civ) B

C




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The quantity N(A)

n can be thought of as arelative frequency. With a

deck of cards we know the quantity N(A), but imagine we did not.

Consider an experiment that can be repeatedly performed in anidentical and independent fashion, and let A be an eventconsisting of a fixed set of outcomes of the experiment.

If the experiment is performed n times, on some of thereplications the event A will occur (the outcome will be in the setA), and on others, A will not occur.

Then, the relative frequency is the number of times a repeatedexperiment yields the event of interest, A, divided by the total

number of repetitions (

N(A)

n ).

This quantity will vary each time the experiment is conducted and

as n gets large, N(A)

n will approach a limiting value known as the

limiting relative frequencywhich we will identify as P(A).




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Interpreting and Computing Probabilities - 4a

For example, let A be the event that a package sent within the stateof California for 2nd day delivery actually arrives within one day. Theresults from sending 10 such packages (the first 10 replications) areas follows:




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Interpreting and Computing Probabilities - 4b




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Additional Question


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