STUDY GUIDE AND WORK BOOK

STAT/MATH 2501

FALL, 2006

ENGIN A. SUNGUR

PROFESSOR OF STATISTICS

 

TABLE OF CONTENTS

I. Chapter 1                                                                                                                                                               1

I.1. Case Study                                                                                                                                                            1

I.2. Historical Remarks                                                                                                                                          1

I.3. Review                                                                                                                                                                   2

Set Theory                                                                                                                                                                  2

Basic Set Operations                                                                                                                                                      3

Additional Definitions on Sets                                                                                                                      3

Simple Laws and Some Extensions on Sets                                                                                                                    3

I.4. Overview                                                                                                                                                              4

Where We're Going                                                                                                                                                  4

Reading Probabilistic Statements                                                                                                                     5

I.5. Technical Details                                                                                                                                             6

Independent                                                                                                                                                               7

How to Solve Problems Related with the Probability?                                                                             9

Case Study Revisited                                                                                                                                             16

I.6. Drunkard's Walk

II. Chapter 2                                                                                                                                                           33

II.1. Review                                                                                                                                                                33

Discrete Mathematics                                                                                                                                        33

Binomial Theorem                                                                                                                                                 33

Geometric Series                                                                                                                                                  33

Maclaurin Series                                                                                                                                                   34

The Derivative of a Function                                                                                                                           34

Derivatives of the Composite Functions                                                                                                    34

The Definite Integral of a Function over an Interval                                                                          35

Antidifferentiation                                                                                                                                             35

Evaluation of the Indefinite Integrals                                                                                                      36

Evaluation of the definite integral by using antiderivatives                                                          36

The Three Elementary Methods of Integration                                                                                      36

Integration by Substitution                                                                                                                            36

Integration by Parts                                                                                                                                         37

Some Results Involving Limits                                                                                                                        37

Some Results Involving Multivariate Calculus                                                                                      38

Some Special Functions                                                                                                                                      40

Exponential Function                                                                                                                                                  40

Gamma Function                                                                                                                                                         41

Beta Function                                                                                                                                                              41

References                                                                                                                                                             43


I. Chapter 1

 

Introduction to Probability Theory

 

 

I.1. Case Study

 

(Motivation: Application of the topics that will be covered in this chapter to the real world problems)

 

Radelet (1981) studied effects of racial characteristics on whether individuals convicted of homicide receive the death penalty. The events that are considered on this study are the selection of a case with "death penalty verdict", "not death penalty verdict", "white defendant", "black defendant", "white victim", and "black victim". The 326 subjects were defendants in homicide indictment in 20 Florida counties during 1976-1977. The following table gives the number of subjects for each of the defendant's race, victim's race and death penalty combinations.

 

                                                                          Death  Penalty

Defendant's Race

Victim's Race

Yes

No

Total

White

White

19

132

151

 

Black

0

9

9

Black

White

11

52

63

 

Black

6

97

103

Total

 

36

290

362

Source: Agresti, A. Categorical Data Analysis, John Wiley & Sons, 1990, pg. 135-138

 

The main question that one would like to answer is "Is there an evidence of racial discrimination given the evidence on this table?". Also, one would be interested with the following questions; (i) is there a relation between defendant's race and victim's race?, (ii) is there a relation between victim's race and death penalty? (iii) If we control for the victim's race, that is if we look at the cases for black victims and white victims separately, what is the relation between defendant's race and death penalty verdict?

 

I.2. Historical Remarks

(Motivation)

 

In the long run, we are all dead.

-John Maynard Keynes (England, 1883-1946)

People talk loosely about chance all the time, without doing any harm. What are the chances of getting a job? of meeting someone? of rain tomorrow? of passing Math. 3610? But for scientific purposes, it is necessary to give the word chance a definite, clear interpretation. This turns out to be hard, and mathematicians have struggled with the job for centuries. They have developed several careful and rigorous theories of chance; but the theories cover just a small range of cases where people ordinarily speak of chance.

 

Exercise 1. How would you explain the fact that when you toss a coin the chance of getting head is one-half? How many possibilities we have if you toss a coin? Can you repeat the experiment, that is tossing a coin, under the same conditions many times?

 

Exercise 2. What is the meaning of "chance of passing Math. 3610 is 0.80"? How many possibilities we have for this experiment? Can you repeat the experiment, that is taking Math. 3610, under the same conditions many times? What could be the conditions that would change if you repeat the experiment?

 

Frequentist definition of chances works best for the processes that can be repeated over and over again, independently under the same conditions. Games of chance fall into this category, and in fact much of the frequency theory was developed to solve gambling problems. One of the great masters was Abraham de Moivre, a French Protestant who fled to England to avoid religious persecution. In his book The Doctrine of Chances he included the following letter,

 

To the Right Honorable the

Lord CARPENTER

 

My Lord

 

There are many people in the World who possessed with an Opinion, that the Doctrine of Chances has a Tendency to promote Play; but they soon will be undeceived... Your Lordship does easily perceive, that this Doctrine is far from encouraging Play, that is rather a Guard against it, by setting in a clear light, the Advantages and Disadvantages of those Games wherein Chance is concerned...

 

Another use to be made of this Doctrine of Chances is that it may serve in conjunction with the other parts of the Mathematicks, as a fit introduction to the Art of Reasoning: it being known by experience that nothing can contribute more to attaining that Art, than the consideration of a long Train of Consequences, rightly deduced from undoubted Principles, of which this Book affords many examples.

 

Historically, probability had its origin in the gambling room. The Chevalier de Mere, a professional French gambler, had asked his friend Blaise Pascal (1623-1662) to solve the following problem: In what proportion should two players of equal skill divide the stakes remaining on the gambling if they are forced to stop before finishing the game? Pascal wrote to and began an active correspondence with Pierre Fermat (1601-1665) concerning the problem (collaboration). Although Pascal and Fermat agreed on the answer, they gave different proofs (nonuniqeness of the solutions). It was in this series of correspondences during the year 1652 that they developed the modern theory of probability.

 

A century earlier the Italian mathematician and gambler (interesting combination!) Girolomo Cardan (1501-1576) wrote The Book on Games of Chances. This is really a complete textbook for gamblers since it contains many tips on how to cheat successfully. Cardan was also an astrologer. According to the legend, he predicted his own death astrologically and to guarantee its accuracy he committed a suicide on that day. (Of course, that is the most convincing way to be right!) Like Cardan, De Moivre predicted the day of his death. A rather interesting story is told of De Moivre's death. De Moivre was ill and each day he noticed that he was sleeping 15 minutes longer than he did on the preceding day. Using progressions, he computed that he would die in his sleep on the day after he slept 23 hours and 45 minutes. On the day following a sleep of 24 hours, De Moivre died.

 

The French mathematician Pierre Simon de Laplace (1749-1827) also contributed much to the historical development of probability. In 1812 he published Theorie Analtique des probabilities, in which  he referred to probability as a science that began with games but that had wide-ranging applications. In particular, he applied probability theory not to gambling situations but as an aid in astronomy.

 

Over the course of many years probability theory has left the gambling rooms and has grown to be an important and ever-expanding.

 

I.3. Review

 

Set Theory

 

In this section we will look at the basic notions and notations needed for dealing with sets (collection of objects of various kinds, such as numbers or paired of numbers, called points). Although you may have dealt with these concepts since grade school, it is important that this chapter be utilized to assure the sound base for the applications of the probability that follow in Chapter 1 of your textbook.

 

"Set" is an undefined notion (like "point" or "line" in high-school geometry); it is assumed that at least one exists, and attention is restricted to a universe or universal set. In probability theory universal set corresponds to the sample space S, sets correspond to the events. All operations on sets are with respect to S in order to avoid paradoxical situations.

 

Exercise 4. The Barber of Seville shaves all men who don't shave themselves. Who shaves the Barber of Seville.

 

 

Basic Set Operations

 

There are four basic set operations which we will illustrate their meanings with diagrams called Venn diagrams.

 

Definition 1. Union (or join, or sum) of sets E and F:

E F E + F {w: w E or w F} (which you read the last notation as E union F consists of points w which are elements of E or F).

 

 

Definition 2. Intersection (or meet, or product) of sets E and F:

E F EF {w: w E and w F}.

 

Definition 3. Complement of set E:

E E' {w: w S and w E}.

 

Definition 4. Difference of sets E and F:

E - F E F {w: w E and F F}.

 

Additional Definitions on Sets

 

Definition 5. E is a subset of F, written E C F, if w E Þ w F.

Definition 6. The empty set (or null set), usually written , is the set containing no points at all.

Definition 7. Sets E and F are said to be mutually exclusive  (or disjoint)  if

E F = .

 

Simple Laws and Some Extensions on Sets

 

In the following theorem we are going to state some simple laws dealing with the operations on sets.

 

Theorem on Sets: Let S be the sample space and E, F, and G be the events on S. Then

1. E E = E, and E E = E (Idempotency Laws).

2. E F = F E, and E F = F E (Commutative Laws).

3. E (F G) = (E F) G, and  E (F G) = (E F) G (Associative Laws).

4. E (F G) = (E F)  (E G) , and E (F  G) = (E F)  (E G) (Distributive Laws).

5.                     E S = E,                               E S = S,      

                        E  = ,                            E = E,

                        E E= ,                           E E= S,

6.  (E F)= E F, and (E F)=  E F.

7. (E)= E

8. F is a subset of E if and only if E F = F.

9. F is a subset of E if and only if E F = E.

Although the terminology given in parentheses is not now in widespread use, it is important to expose to it in order to be able to read other (and older) books and writings.

Exercise 5. Translate the following informal statements and definitions into the formal form by using the set notations.

 

Occurrence of two thing at the same time

 

 

Occurrence of one thing prevents the occurrence of the other

 

 

Occurrence of one thing implies the occurrence of the other

 

 

Occurrence of either one of the two thing

 

 

Occurrence of an opposite of one thing

 

 

Occurrence of three things at the same time

 

 

Occurrence of only one of the two things

 

 

Occurrence of one thing given the occurrence of the other

 

 

Occurrence of any one of the possibilities

 

 

Occurrence of an neither one of the two things

 

 

Occurrence of the either one of the three distinct things

 

 

 

 

 

I.4. Overview

Where We're Going

 

Today, probability theory is a well established branch of mathematics that finds its applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risk of a new medical treatment.

 

Most important of all it will give us the necessary tools to find a measure of reliability for the generalizations that we are going to make from sample to the whole population. For example when you say "Vikings will win the super bowl in 1992-93", is this statement reliable 100%? What is the reliability measure of this statement?

 

To understand the importance of the probability theory better, we should grasp the difference between theory  and reality. But note that reality does not always turn out to be same as theory, that is what we expect. For example, in the theory if you get your graduate degree, you should be able to get a well-paid job. But a person with a graduate education may end up with a low-paid job.

 

 

Theories are ideas proposed to explain phenomena in the real world and, as such, are approximations or models for reality. These models are expressed either in a verbal form or as mathematical relationships. Whereas the theory of social change might be expressed verbally in sociology, the theory of heat transfer is presented in a precise and deterministic mathematical manner in physics. Neither gives an accurate and unerring explanation of the real life. Slight variations from the mathematically expected can be observed in heat transfer phenomena and other areas of physics. The deviations can not be blamed solely on the measuring instruments, the explanation that one often hears, but are due in part to a lack of agreement  between theory and reality. Anyone who believes that the physical scientist completely understands the wonders of this world  need only look at history to find a contradiction. Theories assumed to be the "final" explanation for nature have been superseded in rapid succession during the past century.

 

In this course, we shall develop certain models of reality (probabilistic models); we shall attempt to explain the motivation behind such a development and uses of the resulting model. We will never claim that some statement is always correct, but we will attach reliability, a type of chance statement on the validity of our statement.

 

Reading Probabilistic Statements

 

Exercise 6. Translate the following informal statements and definitions into the formal form by using the notations used in your text. The informal statements are taken from the Doctrine of Chances.

 

 

The chance of something gives the percentage of time it is expected to happen, when the basic process is done over and over again, independently under the same conditions.

 

P(E)=

The chances are between 0% and 100%

 

 

 

The chance of something equals 100% minus the chance of the opposite thing.

 

 

If the occurrence of one thing prevents the occurrence of the other, then the chance of observing these two things together is zero.

 

 

If the chance of occurrence of one thing is not affected by the occurrence of the other thing they are called independent.

 

 

Given the occurrence of one thing the chance of occurrence of another thing is proportional to chance of observing these two together, and inversely proportional to the chance of observing the given thing

 

 

The chance of observing two things at the same time is equal to the chance of observing one of them times the chance of observing other given the one we know its occurrence chance.

 

Two things are independent if the chances for the second one given the first are the same, no matter how the first one turns out.

 

 

The chance of occurrence of one thing is equal to the chance of occurrence of a second thing and that thing together, plus the chance of nonoccurrence of the second thing and that thing together.

 

 

If two things can not occur together than the chance of observing either one of them is the sum of chance of observing each.

 

 

 

 

I.5. Technical Details

 

The following are the basic results that you have learned in this chapter:

 

P(E) = 1 - P(E)

P(E F) = P(E) + P(F) - P(E F) = P(E) + P(F) - P(EF)

P(E | F) =

 

If E and F are mutually exclusive then, P(EF) = 0

If E and F are independent then, P(EF) = P(E)P(F)

If E and F are independent then, P(E | F) = P(E) and P(F | E) = P(F)

 

P(E) = P(EF) + P(EF)

 

Exercise 7. Fill out the following table with the formulas that you would use to find the required probability given the information in the first column.

 

 

 

GIVEN

FIND

GENERAL

MUTUALLY

EXCLUSIVE

INDEPENDENT

P(E), P(F),

P(E F)

 

 

 

P(EF)

P(EF) = P(E) + P(F) - P(E F)

P(EF) = 0

P(EF) = P(E)P(F)

P(E | F), P(F)

 

 

 

P(EF)

 

 

 

P(F | E), P(E)

 

 

 

P(EF)

 

 

 

P(E | F), P(F),

P(E)

 

 

 

P(EF)

 

 

 

P(E | F), P(E | F), P(F)

 

 

 

P(E)

 

 

 

P(E | F), P(E | F), P(F)

 

 

 

P(F | E)

 

 

 

P(E), P(F), P(G), P(EF), P(EG), P(FG), P(EFG)

 

 

 

 

 

P(E F G)

 

 

 

P(EFG), P(FG)

P(E | FG)

 

 

 

 

 

 

P(EFG), P(E)

P(FG | E)

 

 

 

 

 

 

 

 

Independent, Mutually Independent and Pairwise Independent Events

 

Definition: The events E, E, ..., Eare said to be mutually independent if

P(EE... E) = P(E)P(E) ... P(E).

 

Definition: The events E, E, ..., Eare said to be pairwise independent if

P(EE) = P(E)P(E) for all i ¹ j.

 

Definition: The events E, E, ..., Eare said to be independent if for every subset E, E, ..., E, r n, of these events

P(EE... E) = P(E)P(E) ... P(E).

 

Exercise 1: Consider the set of events E, E, E, E. What properties need to be shown to establish that these events are independent?

 

For random events we consider the notions of mutually independence and pairwise independence. The questions of interest are: (i) Are there sets of random events which are pairwise independent but not mutually independent?, (ii) Conversely, are there sets of random events which are mutually independent but not pairwise independent? In the following two examples we will try to answer these questions.

 

Example (pairwise independence does not imply mutual independence): (Berstein, 1928) Suppose that a box contains 4 tickets labeled by 112, 121, 211, and 222. Experiment consists of drawing one ticket at random from the box. Let

                        E= "1 occurs at the first place"

                        E= "1 occurs at the second place"

                        E= "1 occurs at the third place".

It follows that

P(E) = P(E) = P(E) = = ,

P(EE) = P({112}) = = P(E)P(E),

P(EE) = P({121}) = = P(E)P(E), and

P(EE) = P({211}) = = P(E)P(E).

That is the events E, E, Eare pairwise independent. However the event EEE= . Therefore

P(EEE) = 0 ¹ P(E)P(E)P(E) = = .

Exercise 2: A box contains eight tickets, each labeled with a binary number. Two are labeled 111, two are labeled 100, two 010, and two 001. An experiment consists of drawing one ticket at random from the box. Let Ebe the event "the first digit is 1", Ebe the event "the second digit is one", and Ethe event "the third digit is 1". Show that E, E, and Eare pairwise independent but not mutually independent.

 

Example (mutual independence does not imply pairwise independence):

An experiment consists of tossing two different standard dice, white and black. The sample space S of the outcomes consists of all ordered pairs (ij), i,j = 1, 2, ..., 6, that is

S = {11, 12, 13, ..., 66}.

Each point in S has a probability 1/36. Define the following events:

                        E= "first die is 1, 2 or 3"

                        E= "first die is 3, 4 or 5"

                        E= "sum of the faces is 9".

Therefore,

                        EE= {31, 32, 33, 34, 35, 36},

                        EE= {36},

                        EE= {36, 45, 54},

                        EEE= {36}.

It follows that

P(E) = P(E) =  , P(E) = and

P(EEE) = = = P(E)P(E)P(E).

However the events are not pairwise independent, because

P(EE) = ¹ = P(E)P(E),

P(EE) = ¹ = P(E)P(E),

P(EE) = ¹ = P(E)P(E).

Exercise 3: In exercise 2, let us change the number on one ticket from 111 to 110, and the number of another ticket from 100 to 101. Show that E, E, and Eare, this time, mutually independent but not pairwise independent.

 

 

How to Solve Problems Related with the Probability?

 

Most important element in the solution of the probability and statistics problems is the nonuniqueness of the solution method. The following graphical and tabular techniques are aimed to help to understand the problem better:

 

I. Straight forward logical set up,

II. Tables representations,

III. Venn diagrams,

IV. Trees.

 

Straight forward logical set up gives the fastest solution of the problem. But it requires a deep understanding of the definitions, concepts, theorems, and formulas. Last three techniques help to eliminate this difficulty. These techniques help to develop a logical and systematic thinking process on the solution of the problem. Direct, blind-folded use of formulas might endanger logical thinking and generally leads to an incorrect solution of a problem. Whatever technique one prefers, the basic steps are;

 

(i) Define the experiment,

(ii) Define the sample space (simple outcomes of the experiment)

(iii) Define the events that are given in the problem by using capital letters

(iv) For compound events use the set notation that we discussed before

(v) Write down the event that we would like to find the chance of occurrence on

(vi) Select the formula that will give us that probability given the information on the events you defined in (iii) & (iv).

(vii) Use your intuition and prior knowledge on the result and interpret that number. Does it make any sense? Is it what you expected?, If not, did you carry out your calculations correctly?

 

Exercise 8: A softball team has three pitchers, A, B, and C, with winning percentages of 0.4, 0.6, and 0.8, respectively. These pitchers pitch with frequency 2,3, and 5 out of every 10 games, respectively.

(a) What is the probability that the team will win the game?

(b) Suppose that the team won the game what is the probability that A pitched the game?

 

Understanding the problem:

What is the experiment? : Each game in this problem is the experiment

What are the possible outcomes of the experiment? : First of all A may pitch the game or B may pitch the game or C may pitch the game. So let us label each one of these simple events

 

            A = the event that A will pitch the game

            B = the event that B will pitch the game

            C = the event that C will pitch the game

 

Secondly, the team may win the game or lose the game, so let us label these outcomes

 

            W = the event that the team will win the game

            L = the event that the team will lose the game

 

What is the relation between the events? That is are they mutually exclusive or independent?

 

            Since the team cannot win and lose the game at the same time W and L are mutually exclusive.

 

            If we assume that only one pitcher pitches the whole game A, B, and C are also mutually exclusive.

            The questions that you must have at this point are the following:

            Are the events A, W mutually exclusive? If yes, what does it say for the pitcher A?

            Are the events B, L mutually exclusive? If yes, what does it say for the pitcher B?

            Are the events A, W independent? If yes, what does it say for the pitcher A?

            Are the events B, L dependent? If yes, what does it say for the pitcher B?

           

            The translation of the mathematics notation to the daily language plays an important role in understanding and solving problems related with probability. The following table shows the translation of these languages from one another.

 

 

Mathematics

Notation

AW=

 

Interpretation

The intersection of the two sets A and W is empty

 

 

Statistics

 

The sets A and W are mutually exclusive

 

 

English

Symbolic

"Winning the game" and "A pitching the game" can not occur at the same time

 

Interpretation

If A pitches then we know that the team will not win the game. Therefore, A is a terrible pitcher.

 

 

 

Now, fill out the following table for the A and W being independent.

 

 

Mathematics

Notation

 

 

Interpretation

 

 

 

Statistics

 

 

 

 

English

Symbolic

 

 

Interpretation

 

 

 

 

What are the numbers that are given in the statement of the problem?

 

            First of all the problem says A has a winning average 0.4. So A will win 40% of the games. But for A to win the game it should play the game as a pitcher. In other words A will win the game given that he plays the game. Therefore,

 

            P(W | A) = 0.40

            P(W | B) = 0.60

            P(W | C) = 0.80.

           

            Since these pitchers pitch with frequency 2, 3, and 5 out of every 10 games, respectively;

 

            P(A) = 0.20

            P(B) = 0.30

            P(C) = 0.50.

 

            The first part of the problem is asking for P(W). What is the difference between P(W) and any one of the following probabilities P(W | A), P(W | B), and P(W | C)?

 

            The second part of the problem is asking for P(A | W). What is the difference between P(W | A) = 0.40 and P(A | W)?

 

            What do you think that P(A | W) + P(B | W) + P(C | W) will be equal to?

 

            Given the information on P(A), P(B), P(C), P(W | A), P(W | B), and P(W | C), Do you expect that P(A | W) will be greater than P(C | W) ? Why or why not?

 

            Note that the components that will affect the P(A | W) are (i) A should play, that is P(A) and (ii) the team should win, that is P(W | A).

 

            Now, we will solve the problem by using the 4 techniques that we introduced above.

 

            Solution I. To find out P(W), one might use the following logical steps:

(i) In which cases the team will win the game. The team will win the game when A pitches, or B pitches, or C pitches. Therefore

 

"The team wins the game" = "A pitches and the team wins"

                                                            or "B pitches and the team wins"

                                                                        or "C pitches and the team wins".

 

By using our notation;

 

W = (A W) (B W) (C W) = AW  BW    CW

 

Are the events AW, BW, CW mutually exclusive? Justify your answer.

 

P(W) = P(AW  BW    CW) =  P(AW)  + P(BW)  +  P(CW)  (Why?)

 

Do we know the value of P(AW), P(BW), and P(CW)?

 

Can we find P(AW), P(BW), and P(CW) given the information on P(A), P(B), P(C),

P(W | A), P(W | B), and P(W | C)?

 

By using the multiplication rule (definition of the conditional probability), we end up with

 

P(AW) = P(W | A) P(A) = 0.40 0.20 = 0.08

P(BW) = P(W | B) P(B) = 0.60 0.30 = 0.18

P(CW) = P(W | C) P(C) = 0.80 0.50 = 0.40

 

Note that the multiplication rule on P(AW) can be used also as follows:

 

P(AW) = P(A | W)P(W). Why we did not use this form?

 

Is P(AW) = P(A | W)P(W) = P(W | A) P(A) ? Justify your answer.

 

Therefore, P(W) = P(AW)  + P(BW)  +  P(CW)

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

                                    = 0.08 + 0.18 + 0.40 = 0.66

 

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

 

 

 

 

 

 

 

The second part of the problem is asking for P(A | W). By using the definition of the conditional probability, we end up with

 

 

P(A | W) = =

 

 

 

How did we get the second part of the above formula?

 

 

Therefore,

 

P(A | W) = = 0.121. Similarly P(B | W) = 0.273, and P(C | W) = 0.606.

 

We are going to interpret these results after we discuss all of the solution techniques.

 

Solution II. Tables. The information that is given in the problem can be represented by using a table. In this table the rows will show the result of the game, and the columns will show who pitches the game. The elements in the table will provide the probability that the corresponding row and column elements occur together.

 

 

 

 

A PITCHES THE GAME

(A)

B PITCHES THE GAME

(B)

C PITCHES THE GAME

(C)

 

THE TEAM WINS THE GAME

(W)

 

P(AW)

 

P(BW)

 

P(CW)

 

P(W)

THE TEAM LOSES THE GAME

(W)

 

P(AW)

 

P(BW)

 

P(CW)

 

P(W)

 

 

P(A)

 

 

P(B)

 

P(C)

 

 

1

 

Note that this table gives us some important formulas such as

 

P(AW) + P(BW) + P(CW) = P(W)

 

P(AW) + P(AW) + P(AW) = P(W)

 

If the conditional probabilities are available table can be modified by using the multiplication rule as follows;

 

A PITCHES THE GAME

(A)

B PITCHES THE GAME

(B)

C PITCHES THE GAME

(C)

 

THE TEAM WINS THE GAME

(W)

 

P(AW)=P(W|A)P(A)

 

P(BW)=P(W|B)P(B)

 

P(CW)=P(W|C)P(C)

 

P(W)

THE TEAM LOSES THE GAME

(W)

 

P(AW)=

P(W|A)P(A)

 

 

P(BW)=

P(W|B)P(B)

 

P(CW)=

P(W|C)P(C)

 

P(W)

 

 

P(A)

 

 

P(B)

 

P(C)

 

 

1

 

Now, let us fill out the table with the probabilities that are given

 

 

A PITCHES THE GAME

(A)

B PITCHES THE GAME

(B)

C PITCHES THE GAME

(C)

 

THE TEAM WINS THE GAME

(W)

 

0.4 0.2 = 0.08

 

0.6 0.3 = 0.18

 

0.8 0.5 = 0.40

 

P(W)=?=

THE TEAM LOSES THE GAME

(W)

 

P(W|A)P(A)

 

 

P(W|B)P(B)

 

P(W|C)P(C)

 

P(W)=?=

 

 

0.20

 

 

0.30

 

0.50

 

 

1

 

Fill out the rest of the table, and find P(W) and P(A | W).

 

 

 

 

Solution III. This time we will use a Venn diagram. Each region on the diagram will correspond to a relevant event.

 

Since the probability is a special type of measure, just like the area of each piece on the graph, same type of argument that is given in the solution I can be used to answer to the question.

 

Solution IV. Trees. When we analyze an experiment that takes place in a sequence of steps, we often find it convenient to represent the sample space by the set of all paths through the tree. A probability measure is assigned by considering the conditional probabilities appropriate for the outcome of the any step, given all previous outcomes. These weights are assigned at the appropriate branches of the tree, and then the weights for a path through the tree is the product of the branch weights along the path. In our experiment first of all one of the pitchers should pitch the game then the team either will win or lose. Therefore corresponding tree can be constructed as follows:

 

 

Mark the paths on the tree where the team ends up winning the game. By using these paths write down the formula for finding P(W)

 

 

P(W) =

 

 

Find the probability that the team will win the game by using the above formula

 

 

P(W) =

 

 

 

Suppose that the team wins the game. Given this information some of the branches of the tree will be chopped off.

 

 

Write down the formula to find P(A | W)

 

 

P(A | W) =

 

 

 

Use the formula to find P(A | W)

 

 

P(A | W) =

 

 

 

 

 

Our original tree gave us the probabilities for the result of the game given the pitcher. The i.inverse probabilities,; that is probabilities for the pitcher given the result of the game are called Bayes probabilities. These inverse probabilities can also be obtained by simply constructing the three for the two-stage experiment carried out in reverse order.

 

 

Fill out the rest of the above reverse tree by using P(W) and P(W) that you have already found.

 

Interpretation of the Results

 

In this part we will try to interpret the results that we got. Note that if the pitcher C pitched all the games the probability of winning would be 0.80. The probability of winning reduces to 0.66 since all the games are not pitched by C and all the pitcher are not equally good.

 

If all the pitchers are equally good, say with the winning percentage 0.80, what would be the probability that the team will win?

 

The winning percentages for the pitchers A, B, and C are 0.40, 0.60, and 0.80 respectively. And A, B, C pitches 20%, 30%, and 50% of the games.

 

Given the winning percentages for each pitcher, is it reasonable if A, B, and C pitches 50%, 30%, and 20% of the games?

 

If the team is winning the probability that A is pitching the game is 0.121. If all the pitchers were equally good, that is winning is independent to the whether A is pitching or not, what would this probability be equal to?

 

What type of relation do you expect between P(C | W) and P(C)? (P(C | W) > P(C) or

P(C | W) < P(C) or P(C | W) = P(C))

 

 

 

Case Study Revisited

 

Now we are ready to answer the questions that we rose in our case study. The table of frequencies can be changed into empirical probabilities as follows. For example let us try to find the probability of selecting a case which involves white defendant, white victim, and a death penalty verdict. There are 19 cases out of 326 in which all of these events occurred together. Therefore the chance of observing this event is 19/326=0.058.

 

Define the events of interest for the case study.

 

A = "receiving a death penalty verdict"

B = A=

C = "white defendant"

D = C=

E = "white victim"

F = E=

 

What is the meaning of A and C being mutually exclusive?

 

What is the meaning of A and C being independent?

 

What is the meaning of C and Ebeing mutually exclusive?

 

What is the meaning of C and E being independent?

 

If there is no discrimination, what type of relation you expect to see between A and C? That is, do they have to be mutually exclusive, or independent, or both?

 

Which one of the following probabilities will give you a better information on discrimination? P(A | C) or P(C | A). Similarly P(A | D) or P(D | A).

 

Construct the marginal table for defendant's race and the death penalty verdict by ignoring victim's race.

 

                               Death  Penalty

Defendant's Race

Yes

No

Total

 

White

 

 

 

 

Black

 

 

 

 

Total

 

 

 

 

 

Find

 

P(A)

 

P(A | C) =

 

P(A | D) =

 

P(C | A) =

 

P(D | A) =

 

Is there an evidence of racial discrimination?

 

Construct the same table by controlling for the victim's race. That is create a table for defendant's race and death penalty verdict for white victims and black victims.

 

 

 

For white victims

                              Death  Penalty

Defendant's Race

Yes

No

Total

 

White

 

 

 

 

Black

 

 

 

 

Total

 

 

 

 

 

 

 

For black victims

                               Death  Penalty

Defendant's Race

Yes

No

Total

 

White

 

 

 

 

Black

 

 

 

 

Total

 

 

 

 

 

Find

 

P(A | CE) = P("death penalty" | "white victim" and "white defendant') =

 

P(A | DE) =

 

P(A | CF) =

 

P(A | DF) =

 

Is there an evidence of racial discrimination when we control for the race of the victim?

 

 

Construct a marginal table for the defendant's race and victim's race.

 

Death  Penalty

Defendant's Race

Yes

No

Total

 

White

 

 

 

 

Black

 

 

 

 

Total

 

 

 

 

 

Find

 

P(E | C) =

 

P(E | D) =

 

P(F | C) =

 

P(F | D) =

 

What type of relation we have between defendant's race and victim's race?

 

 

 

 

 

 

 

 

 

 

I.6. Drunkard's Walk:

 

Random Walk:

Where is a drunk person going to end up?

Motivation: Scenario

Most of the problems we encounter in daily living may be modeled probabilistically. The probability models can help us to understand the nature of the problems that we are facing better. Elementary probability models assume that the possible outcomes for each experiment are the same and the outcome of one experiment does not influence the outcomes of the other experiments probabilistically. In some cases these assumptions are not valid. For example;

* how much money you are going to have on your saving account next year depends on how much money you have on this account this year,

* next years interest and unemployment rates will be affected by the this years rates,

* a gamblers fortune at a game depends on the gamblers fortune at the previous game (Gamblers Ruin Problem)

* the population size of an endangered species in year 2000 depends on the population of this species in previous years.

Big Question

Which leads to the big question: How do we model a phenomenon where knowledge of the outcome of previous experiments influences our predictions for the outcomes of the next experiment?

 

            Learning Objectives

Learn how to model chance processes for which the knowledge of previous outcomes influences prediction for future experiment. After completing this activity you should be able to construct a probability model for the cases where outcome in the future given the outcomes in the past and present depends only on the outcome that we observe in the present. You should also be able to see how different tools, such as tree diagrams, graphs, matrices, and tables, can be used to simplify the problem.

 

Background: Review, Definitions, Facts & Concepts

We will be learning Markov chains that are a type of a stochastic process. A stochastic process is a squenceof random variables. One can represent a stochastic process as {X(t), tT} where for each tT, X(t) is a random variable. The index t is often interpreted as time and, as a result, we refer to X(t) as the state of the process at time t. T is called the index set of the process. If the index set is countable, the stochastic process is a discrete-time process. If T is an interval of the real line, the stochastic process is said to be a continuous-time process. Generally, the discrete-time process is represented by Xn instead of X(t). S, state space, is the set of all possible random variables, X(t), of the stochastic process. A realization of a stochastic process is called a sample path of the process.

As warm-up Activity, consider the Mouse in a Maze. A hungry mouse starts in cell 1, while a cat is hiding patiently in cell 7, and the cheese is in cell 9. The mouse moves from cell to cell. . In the absence of learning, it will choose the next cell to visit randomly from the adjoining cells. Assume that once the mouse finds the piece of cheese or the cat, it will understandably stay there forever.

 


 

 



Here is a sample path for this example:

 

Step (n)

0

1

2

3

Xn

1

2

3

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Step (n)

4

5

6

7

Xn

5

4

5

8

 


 


Step (n)

8

Xn

7

 


 


What do you understand when one says

X0=1

X5=4

X8=7

 

 

What is the state space S of this process?

 

First of all, why is this a stochastic process?

 

 

 

Study the following matrix; it is called a one-step transition probability matrix.  What ideas do you have about how this matrix was constructed and what it means?

 

 

Study the following matrix; it is called an incidence matrix.  What ideas do you have about how this matrix was constructed and what it means?

 

 

Study the following graph; it is called a state diagram.  What ideas do you have about how this graph was constructed and what it means?

 


 

 

 


The following matrix is obtained by multiplying the matrix I by itself. If you do not understand how to multiply two matrices do not worry. Just tell us what you think that this matrix is representing.

 

 

 

The following matrix is obtained by multiplying the matrix I by itself three times. What do you think that this matrix representing?

 

 

 

 

 

 

 

 

 

 

 

 

The following matrix is obtained by multiplying the matrix I by itself four times. What do you think that this matrix representing?

 

 

 

Suppose that the cat is on vacation and you are given the following incidence matrix.

 

   

 

Draw the maze (some of the doors between the cells are now closed)

Tell us where the cheese is.

 

 

 

 

 

 

 

 


 


Complete the following state diagram.


 

 


Do you think that drawing the above graph would help you to answer the last two questions?

 

 

 

 

 

 

 

Now, let us challenge ourselves. (I do not have slightest idea at this point whether this can be solved or not) This time it is the Cheeses turn to take a vacation. You are given the following two-step incidence matrix.

 

 

 

 

 

Draw the maze.

Find where the cat is.

 


 

 

 


Learning Activity

Suppose that a person is walking along a four-block stretch of Park Avenue. The person starts at corner x and with probability 1/2, walks one block to the right and, with probability 1/2, walks one block to the left, when the person comes to the next corner she/he again randomly chooses her/his direction. The person continues until she/he reaches corner 4, which is a bar, or corner 0, which is a home. If the person reaches either home or the bar, she/he stays there. Here is a graphical representation of the problem.

 


1.         You will need a coin or other method of simulating an event with a probability 1/2. Pick a starting point 1, 2, or 3. Toss a coin. If the coin shows head move to the right, if it is tail move to the left. Continue tossing the coin until you reach to the bar or the home.

            For example, if you get the following sequence and start at 1: H H T H T

Your position at each step will be: 2 3 2 3 2.

Since you could not reach the bar of the home you should continue tossing the coin.

2.         a.         Starting from 1, repeat the above experiment 10 times, fill out the following table, and record the number of times you ended up home and bar.

Coin Toss Sequence

Position

Destination

H H T H T T T

2 3 2 3 2 1 0

Home

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Total Number of homes

 

 

 

b.         Starting from 2, repeat the above experiment 10 times, fill out the following table, and record the number of times you ended up home and bar.

Coin Toss Sequence

Position

Destination

H H T H T T T

3444444

Bar

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Total Number of homes

 

 

c.         Estimate you chance of reaching home starting from 1 by using the results of the experiments? What about your chance of reaching the bar?

            Estimate you chance of reaching home starting from 2 by using the results of the experiments? What about your chance of reaching the bar?

            Compare your results and justify why these estimates are different.

Connections

3.         Tree Diagrams

a.         Consider the following tree which represents all possible outcomes of tossing a coin 4 times. Follow the individual branches and determine where the drunk person will be if she/he starts at 3.

b.         Since, if the drunk person reaches either home or the bar, she/he stays there, eliminate the unnecessary branches of the tree diagram given above.

c.         Is it possible that starting from block 3 this drunk person will never be able to reach home or the bar? Give an example.

 

4.         From Trees to Matrices

As you can see representing this problem in a compact form by using the trees is difficult. You have to be able to produce tree diagrams with infinite number of branches for each starting point. Now let us search for a better way of summarizing this problem.

a.         Suppose that we saw this person at one of the blocks, the person had enough time to take only one step. Where this person will be when we come back? What are the factors that will affect your answer?


b.


 


c.         Fill out the following table cross out the impossible steps, for the remaining ones put down the corresponding outcome of the coin toss experiment.

                                      Future Position (Position after one step)

 

Present

Position

0

1

2

3

4

0

 

 

 

 

 

1

 

 

 

 

 

2

 

 

 

 

 

3

 

 

 

 

 

4

 

 

 

 

 

 

            Replace corresponding outcomes of the experiment with the probability that matches. Now, we can represent the table of probabilities in a compact form as follows:

 

 

            Interpret this matrix by referring to the table in c. This matrix is called one-step transition matrix. The states 0 and 4 are impossible to leave such states are called absorbing states.  States which are not absorbing called transient.

Reaction and Reflection

5.         Markov Property

 

            Suppose that the drunk person was in block 1, then 2, then 1, then 2 and now the person is in block 3. You want to know where the person will be after the next step. Do you need to know that the person was in corners 1, 2, 1, 2, to determine where the person will be after visiting block 3?

            Suppose that the present the person is in block 3, does it make any difference if this is the persons first, second, or tenth visit of block 3, to determine where the person will be after the next step?

 

 

6.         What will Happen After Two or More Steps?

 

            Suppose that the drunk person was in block 1, where this person could be after two steps? Construct a tree diagram to show possibilities. As an example we have given below a tree diagram when the drunk person was in block 1.


 

 

 


Fill out the following table cross out the impossible steps, for the remaining ones put down the corresponding outcome of the coin toss experiment. Note that this time we have to toss the coin two times.

                                      Position after two step

 

Present

Position

0

1

2

3

4

0

 

 

 

 

 

1

 

 

 

 

 

2

TT

-

TH or HT

-

HH

3

 

 

 

 

 

4

 

 

 

 

 

 

            Replace corresponding outcomes of the experiment with the probability that matches. Now, we can represent the table of probabilities in a compact form as follows:

 

 

Interpret this matrix  which is called two-step transition matrix.  Verify that

 

 

In fact if you want to know what will happen probabilistically after n steps, n-step transition matrix will be:

 

 

            By using a calculator or MINITAB find the 4-step transition matrix and interpret.

MTB > read c1-c5

 DATA> 1 0 0 0 0

 DATA> 0.5 0 0.5 0 0

 DATA> 0 0.5 0 0.5 0

 DATA> 0 0 0.5 0 0.5

 DATA> 0 0 0 0 1

 DATA> end

       5 ROWS READ

 MTB > copy c1-c5 m1

 MTB > print m1

  MATRIX M1

   1.0   0.0   0.0   0.0   0.0

   0.5   0.0   0.5   0.0   0.0

   0.0   0.5   0.0   0.5   0.0

   0.0   0.0   0.5   0.0   0.5

   0.0   0.0   0.0   0.0   1.0

 MTB > multiply m1 m1 m2

 MTB > print m2

  MATRIX M2

   1.00   0.00   0.00   0.00   0.00

   0.50   0.25   0.00   0.25   0.00

   0.25   0.00   0.50   0.00   0.25

   0.00   0.25   0.00   0.25   0.50

   0.00   0.00   0.00   0.00   1.00

 

 

 

 

MTB > multiply m1 m2 m3

 MTB > print m3

  MATRIX M3

   1.000   0.000   0.000   0.000   0.000

   0.625   0.000   0.250   0.000   0.125

   0.250   0.250   0.000   0.250   0.250

   0.125   0.000   0.250   0.000   0.625

   0.000   0.000   0.000   0.000   1.000

MTB > multiply m1 m3 m4

 MTB > print m4

  MATRIX M4

   1.000   0.000   0.000   0.000   0.000

   0.625   0.125   0.000   0.125   0.125

   0.375   0.000   0.250   0.000   0.375

   0.125   0.125   0.000   0.125   0.625

   0.000   0.000   0.000   0.000   1.000

 

 

Ongoing Assessment and Wrap-up

7.         Discuss the implications of Markov Properties. List down possible tools that can be used to model a problem. Which one of these tools is the most powerful in your opinion? Why?

8.         Give some examples where this process can be used in the real world.

 

 

Extensions

9.         (Forecasting the weather) The chance of rain tomorrow depend on only todays whether. If it rains today, then it will rain tomorrow with probability 0.7 and if it does not rain today then it will rain tomorrow with probability 0.4. Form the one-step transition matrix.

 

            Find the two-step transition probability matrix. What is the probability that it will rain two days from today given that it is remaining today?

            Find the four-step transition probability matrix and interpret. Did you notice any interesting pattern in probabilities?

Answers:

 

 

 

 

10.       (War of long distance companies) It has been observed that customers switch from one long distance company to another according to the following transition matrix.

 

 

            Find the two-step transition matrix and interpret one and two-step transition matrices. Find the probability that a customer who uses MCI today will again be using MCI two months from today, assuming that she or he switches once a month.

 

 

            Find the twelve step transition matrix. Interpret the matrix.

Answer:

 

11.       (Gossip spreading) If a member of a community is entrusted with some news, she/he will tell it to her/his friends truly and without any changes with probability p. The probability that she/he will twist it around and tell the opposite is really negligible, say, . The probability of such a misconduct and misbehavior is less than one in a billion. Therefore, if a member of the community receives news, she/he will tell it to another member truly and unchanged with probability p, and she/he will twist it and tell its opposite with probability . Let

T= the true statement about the young persons dog

N= the negation of the true statement about the persons dog.

Construct the one-step transition matrix. Interpret the matrix. Find ten, twenty, one hundred-step transition probability matrices. Do you see anything interesting? How are the probabilities changing? Can you say that in general in this community only half of the people is telling the truth?

 

 

12. Visit the Mouse in a Maze example and place the cat so that the chance that the mouse will end up with cheese will be lower than the cat catching the mouse.
I.7. A Markov Chain Activity

 

 

MARKOV CHAINS

 

Objective: To learn how to set up probability transition matrix and construct an appropriate state diagram.

 

Activity:

1. Consider a farmer using an old tractor. The tractor is often in the repair shop but it always takes only one day to get it running again. The first day out of the shop it always works but on any given day thereafter, independently of its previous history, there is a 10% chance of it not working and thus being sent back to the shop.

 

Let X0, X1, ... be random variables denoting the daily condition of the tractor, where a one denotes the working condition and a zero denotes the failed condition.

 

Explain in words

Xn=1

Xn=0

Write one step transition probability matrix

 

P=   

Here is the state diagram for this Markov chain:

 

2. A salesperson lives in town a and is responsible for towns a, b, and c. Each week he is required to visit a different town. When s/he is in her/his home town, it makes no difference which town s/he visits next so s/he flips a coin and if it heads s/he goes to b and if tails s/he goes to c. However, after spending a week away from home s/he has a slight preference for going home so when s/he is in ether towns b or c s/he flips two coins. If two heads occur, then s/he goes to the other town; otherwise s/he goes to a.

 

Define

X0=

X1=

X16=

 

 

Explain in words

Xn=a

Xn=b

Xn=c

 

Write one step transition probability matrix

 

P=   

Construct a state diagram for this Markov chain:

 

 

3. Let {Xn; n=0, 1, ...} be a Markov chain with state space {1, 2, 3, 4} and transition probabilities given by

P=   

 

Construct a state diagram for this Markov chain:

 

 

Starting from state 4 is it possible not to come back state 4 again? Give an example.

What happens if you start with state 1.

 

4. This example illustrates that the n need not refer to time. Consider a page of text and represent vowels by zeroes and consonants by ones. Thus the page becomes a string of zeros and ones. It has been indicated that the sequence of vowels and consonants in the Samoan language forms a Markov chain, where a vowel always follows a consonant and a vowel follows another vowel with a probability of 0.51.

Write one step transition probability matrix

 

P=   

 

ASSESSMENT

 

Each morning an individual leaves her house and goes for a run. She is equally likely to leave either from her front or back door. Upon leaving the house, she chooses a pair of running shoes (or goes running barefoot if there are no shoes at the door from which she departed). On her return she is equally likely to enter, and leave her running shoes, either by the front or back door. Suppose that she owns a total of 5 pairs of running shoes.

Let  Xn denote the number of pairs of shoes at the door the runner departs from at the beginning of day n.

 

Write one step transition probability matrix

 

P=   

 

 

 


 

II. Chapter 2

 

Random Variables

 

II.1. Review

 

For this chapter we need to know some mathematical facts on the functions. These facts will help us to understand the properties of the probabilistic models that we are going to develop. A function f from a set D to a set E is a correspondence that assigns to each element  of D a unique element of E.

 

Exercise 1. Suppose that coin tossed three times, leading to a sample space S={HHH,HTT,THT,TTH,THH,HTH,HHT,TTT}. Define X=Number of heads observed when we tossed a coin three times. Is X a function? If yes, what is D and E?

 

Discrete Mathematics

To be able to understand and derive the properties of the discrete probabilistic models we need the following facts from discrete mathematics.

 

Binomial Theorem;: The quantity (a+b)can be expressed in the form

(a+b)= ,

where

= ; i=0 , ..., n.

For example, we have

(a+b)= 1

(a+b)= a+b

(a+b)= a+2ab+b

(a+b)= a+3ab+3ab+b.

Note: In probability we will use binomial theorem not to extend (a+b)but to evaluate the sums of the form . For example can you figure out the value of the following sum?

 

= ? =

Hint: a=?=       , b=?=

 

What about the following sum?

 

= ? =

Maybe we should look for a better way of evaluating this sum?

 

Geometric Series;:

Let a0.

= a+ar+ar+ar+...+ar+...

is called geometric series which

(i) converges and has the sum if |r|<1.

(ii) diverges if |r|1.

In probability geometric series is used in a discrete distribution which is called geometric distribution. Can you find the value of the following sum?

 

= ? =

Note that the lower limit of the sum starts from 1 not 0 as in the geometric series. Maybe we should define i=n-1. Then, what is a=?=      , r=?=          

 

Maclaurin Series;:

One of the Maclaurian series that is used in the probability is the following

e= = 1+ a + + + ... .

By using the Maclaurian series can you find the value of the following sum?

 

 e= ? =

 

What about the following sum which is used to find expected value of a Poisson random variable.

 

i e= ? =

Note that = , so take one of the lout of the summation and ...

 

The Derivative of a Function:

One way of understanding the behavior of a function f(x), is to look at its derivative. With each function f(x) we associate another function, called "the derivative of f(x)". This derivative is a formula for the rate at which f(x) is changing as x changes, and it measures the steepness of the graph of f(x). Here are the some of the formulas for the derivatives that you should know(g(x) and h(x) are differentiable functions and a, and b are arbitrary constants):

 

function

f(x)

its derivative

f=

ax+b

a

ax+b

nax

e+b

ae

e+b

-ae

e+b

ge

aln(x)+b

h(x)g(x)

hg(x)+h(x)g

Table 1. Table of Derivatives for Some Functions.

 

A derivative of an integral as a function of its limit of integration can be found by using the following relation (f(x) is a "nicely behaved" function)

 

= f(x).

 

Derivatives of the Composite Functions: The Chain Rule

If f=f(u) is a function of input variable u and u=u(x) is a function of the input variable x, then

f(u(x)) = .

This rule is useful in finding derivatives of functions with a complicated relationship.

 

Exercise 2. Find the derivatives of the following functions

(a) F(x)=1 - le

(b) M(t)=

(c) M(t)=

(d) F(x) = dz.

The Definite Integral of a Function over an Interval: The definite integral of a function over an interval, say axb, of x-values, is a number which measures the amount of f(x) aggregated in that interval.

Facts:

(i) If f(a) exists, then = 0.

(ii) If f is integrable and f(x)0 for every x in [a,b], then the area A of the region under the graph of f from a to b is

A = .

(iii) For a < c < b

= +

 

(iv) If f(x)0 and integrable on [a,b], then

0.

 

Antidifferentiation; is the reverse process to differentiation, that is antidifferentiation is the process of finding a formula for a quantity if you its rate of change. The process of antidifferentiation is commonly called integration or indefinite integration. The absence of the values bounding x at the top and bottom of the integration symbol indicate that it calls for antiderivative and not the definite integral.

 

= uv -

= u+ C, n-1

= ln|u| + C

du = e+ C

= a+ C

= tan+ C

= ln + C

= ln + C

Table 2. Table of Integrals

 

Evaluation of the Indefinite Integrals;: By reversing the table of derivatives (Table.1), we obtain a list of antiderivatives of many basic types of functions. More complicated functions need to be re-expressed in a simpler form by algebraic manipulations or substitutions.

 

Evaluation of the definite integral by using antiderivatives; can be done by taking the difference in the values of an antiderivative of f(x) (if one can be determined) at x=b and x=a. That is

= F(x)F(b) - F(a)

where,

F(x) = .

Note that for the definite integral the constant in the antiderivative disappears.

 

The Three Elementary Methods of Integration;. Given an integral to evaluate whose formula is not in the table of integrals: what to do? Often we can obtain the result we seek by using one of the three elementary methods of integration from the calculus. Do you remember them? 1) Ask your room-mate; 2) ask the instructor; and 3) look it up. That is only partly facetious. There are extensive tables of integrals in the CRC-Handbook and in other sources, and "Mathematica" is a powerful piece of computer software that will readily evaluate integrals far more difficult than anything that we are likely to encounter. Nonetheless, the three elementary methods are useful tools with which we should be familiar. They are 1) integration by substitution; 2) integration by parts; and 3) integration by partial fractions. In this course we will mostly use the first two methods which are explained below.

      

Integration by Substitution;.

Indefinite Integral

I will explain the method for the following trivial integral,

 

Step 1. Find a u=u(x) such that when we plug in u the unfamiliar integral will transform into one we recognize.

Step 2. Find the relation between dx and du, and replace former by the later.

Step 3. Find the integral (antiderivative) of the recognizable form.

Step 4. Plug in u(x) in your result.

 

We know that = ln|u| + C, so reasonable choice for

(recognize u(x))

u(x) = 3x+5, and

(relation between dx and du)

du=3dx Þ dx= du

(replace former by the later, and find antiderivative)

du = = ln|u| + C

(put back u(x))

= ln|3x+5| + C

 

 

Definite Integral

I will explain the method for the following trivial integral,

 

Step 1. Find a u=u(x) such that when we plug in u the unfamiliar integral will transform into one we recognize.

Step 2. Find the relation between dx and du, and replace former by the later.

Step 3. Find the new limits in terms of u.

Step 4. Find the integral (antiderivative) of the recognizable form.

Step 5. Find the value of the definite integral.

 

We know that = ln|u| + C, so reasonable choice for

(recognize u(x))

u(x) = 3x+5, and

(relation between dx and du)

du=3dx Þ dx= du

(replace former by the later, find new limits, and find antiderivative)

=

du = = {ln11 - ln5} = ln

 

 

 

Integration by Parts;.

(Attention: This is the method that we are going to use most frequently)

This method is based on the simple rule for finding the differential of a product:

                                                         d(uv) = vdu + udv

Rearranging and integrating,

                                                         = uv -

it being understood that if these are not definite integrals, we must add an integration constant. We are initially stumped by the integral on the left, but in our mind's eye we see a way to rearrange the integral into two parts such that we can evaluate the integral on the right.

 

Here is an example and the steps that you need for this method:

integral

dx

define u and dv

u=x and dv=edx

(Note that we know how to find the antiderivative of dv)

find du and v

du=1dx and v== e

plug in u, du, v, and dv into the formula for integration by parts

= uv -

dx = xe- dx

and the result

dx = xe- e+ C

 

For some problems it is necessary to repeatedly use integration by parts. For such type of problems it is more useful to produce reduction formulae. For example using integration by parts we can establish that

 

= xe- dx

This formula can be used repeatedly until all powers of x have been removed.

 

Some Results Involving Limits

In this section, we simply remind you of some of the techniques on the calculations of the limits that we find most useful in probability theory. These techniques will be useful on understanding Central Limit Theorem, and finding moments by using moment generating functions.

 

Early in a calculus course the existence of the following limit is discussed and is denoted by the letter e:

 

e = lim= lim

 

 

Often it is rather easy to see the value of certain limits. For example, with -1 < r < 1, the sum of the geometric progression allows us to write

 

lim= lim= .

 

Since limr= 0, the limit of the ratio can be easily found.

 

However it is not that easy to determine the limit of every ratio; for example consider

lim= lim.

Since both the numerator and the denominator of the latter ratio are unbounded, we will not be able to find the limit easily.

 

Consider the ratio f(x)/g(x), if the limits of f(x) and g(x) have the limits or - as x approaches to c, we say that f(x)/g(x) has the indeterminate form / at x=c. If the limits of both functions approaches to 0, then we say that f(x)/g(x) has the indeterminate form 0/0 at x=c. L'Hospital's Rule can be applied to the indeterminate forms to find the limit of the ratio.

 

L'Hopital's Rule: Suppose the functions f and g are differentiable on an open interval (a,b) containing c, except possibly at c itself. If g'(x) 0 for x c, and if f(x)/g(x) has the intermediate form 0/0 or / at x=c, then

 

lim= lim

provided that lim[f'(x)/g'(x)] exists or lim[f'(x)/g'(x)] = .

 

 

Therefore, limcan be found taking the limit of the ratio of the derivative of the numerator and the derivative of the denominator. We have

 

lim= lim= 0.

Note that you can use L'Hopital's rule more than once.

 

Some Results Involving Multivariate Calculus

In this section we only make some suggestions about functions of two variables, say

z = g(x

But these results can be extended to more than two variables. Usually all the results that we have learned for the one variable case can be used for the two variable case by treating the "other" variable as a constant. The two first partial derivatives; with respect to x and y, denoted by and  can be found in the usual manner of differentiating by treating the "other" variable constant. For illustration consider the following simple examples,

z = f(x,y) = xy

 

= = = y

and

= = (e)(2xy).

 

The second partial derivatives are simply first partial derivatives of the first partial derivatives. If z=xy, then

= = 1.

 

For notation we use

= ,

= ,

 = ,

= .

In general,

 = ,

provided that partial derivatives involved are continuous functions.

 

The value of the double integral

 

can usually be evaluated by an iterated procedure; that is, evaluating two successive single integrals. For illustration, say A = {(x,y): 0 x 1, 0 y x}.

Then

 

When placing the limits on the iterated integral, note that for each fixed x between zero and one, y is restricted to the interval zero to x (see the figure below). Also in the inner integral on y, x is treated as a constant.

      

In evaluating this double integral we could have restricted y to the interval zero to one, then x would be between y and one. That is, we would have evaluated the iterated integral

Finally, we will look at change of variables in a double integral

If f(x,x) is a joint probability density function of Xand X, then the above double integral represents P[(X,X)A]. Consider only one-to-one transformations, say y=gand y=gwith inverse transformations given by x=hand x=h. The determinant of order 2

is called the Jacobian of the transformation;. And

is called the Jacobian of the inverse transformation. Moreover, say the region A maps onto B in the (y,y) space. Since we are usually dealing with probabilities in this course, we fixed the sign of the integral so that it is positive by using the absolute value of the Jacobian. Then it is true that

That is P[(X,X)A] = P[(Y,Y)B].

 

Note that |J(y,y)| = |J(x,x)|, but the right side is often easy to obtain, rather than solving for xand xin terms of yand yor differentiating implicitly.

 

Now, let us look at the steps required for the multivariate change of variables on the integration.

 

f(x,x)

DETERMINE THE REGION A

DETERMINE gAND g

DETERMINE hAND h

DETERMINE THE REGION B

FIND THE JACOBIAN, J

FIND THE ABSOLUTE VALUE OF THE JACOBIAN, |J|

 

WRITE DOWN THE TRANSFORMED INTEGRAL

 

 

Some Special Functions

Note that our objective is to develop probabilistic models for the characteristic that we are interested with, such as survival time of a cancer patient, life time of a TV set, or simply the midterm exam scores of Math. 3610 students. Since the behavior of all the functions are not the same the functional form of the model will characterize the random variable differently. We will discuss this in detail in the next section. Now let us look at some special functions and their properties.

 

Exponential Function

The function f(x)=eis called exponential function (frequently written exp(x)).

Figure 1. Graphs of eand e.

Note that e=1 and e=0.

ee=e

(e)=e

 

Gamma Function

The gamma function, denoted by G(a) for all a>0, is given by

G(a) =

The gamma function satisfies the following properties:

G(a) = (a-1)G(a-1)   a>1

G(n) = (n-1)! = (n-1)(n-2)...(1)      n = 1, 2, ...

(remember that O!=1 so G(1)=1)

G( ) = .

 

 

Beta Function

For a>0 and b>0, the beta function is given by

beta(a,b) =

The beta function has the following properties:

beta(a,b) =

beta(a,b) = beta(b,a)

 

Exercise 3.

The following type of problems which involves integration in distribution theory to develop probabilistic models:

 

(a) Find the value of k such that

 

= 1.

Answer: k>0.

(b) Find the value of c such that

= 1.

 

Answer: c=12.

 

(c) Find the value of c such that

 

= 1.

Hint: Use the definition of the beta function.

Answer: 20.

 

The following integrals arises in finding distribution functions given probability density functions.

 

(d) Find the value of the following integral

F(x) = .

(F(x) is called cumulative distribution function.)

Answer: 1-x; 1<x<.

 

(e) Find the value of the following integral

F(x) = .

Answer: 1 - e; x>0.

 

The following integrals arises when we want to find expected values.

 

(f) Find the value of the following integral

E(X) = .

(The value of this integral is known as the first moment or expected value of X)

Answer: .

 

 

 

 

(g) Find the value of the integral

 E(X) = .

Answer: .

 

The following type of integrals arises when we want to find the moment generating functions.

 

(h) Evaluate the following integral

 

f= ;  l>0

(fis called moment generating function of exponential distribution with parameter l.)

Answer: .

 

(i) Evaluate the following integral

 

f= .

(fis the moment generating function of a uniform random variable over the interval (a,b)).

Answer: .

 

(j) Find the derivative of the fin (i) with respect to t.

Answer:

 

(k) Find ffor (j).

 

 

(l) Let y=x-xand y=x+x. Find J(x,x)=J, and J(y,y)

Answer: J(y,y)=1/2, J(x,x)=2

 

 

References

 

Agresti, A. Categorical Data Analysis, John Wiley & Sons, 1990

Christensen, H. B., Introduction to Statistics, HBJ, 1992

Dudewicz, E. J., and Mishra, S. N., Modern Mathematical Statistics, John Wiley & Sons, 1988     

Engelhardt, B., Introduction to Probability and Mathematical Statistics, PWS-KENT Publishing Company, 1992

Hogg, R.V., and Tanis, E. A., Probability and Statistical Inference, MacMillan Publishing Company, 1993

Mendenhall, W., Wackerly, D. D., and Scheaffer, Mathematical Statistics with Applications, PWS-KENT, 1990

Sceaffer, R. L., Introduction to Probability, PWS-KENT, 1990


INDEX

Additional Definitions on Sets, 3

antiderivative, 36

Antidifferentiation, 35

Associative Laws, 3

Basic Set Operations TC  "Basic Set Operations" \\l 4 , 3

Bayes probabilities., 15

Beta Function TC  "Beta Function" \\l 4 , 41

Binomial Theorem, 33

change of variables in a double integral, 39

Commutative Laws, 3

Complement, 3

conditional probability, 12

cumulative distribution function, 42

definite integral, 35, 36

derivative, 34

Derivatives of the Composite Functions, 34

determinant, 39

Difference, 3

Discrete Mathematics TC  "Discrete Mathematics" \\l 3 , 33

disjoint, 3

distribution functions, 42

Distributive Laws, 3

double integral, 39

empty set, 3

Evaluation of the definite integral by using antiderivatives, 36

Evaluation of the Indefinite Integrals, 36

event, 2

expected values, 42

exponential distribution, 42

exponential function, 40

first .i.partial derivatives, 38

gamma function, 41

geometric distribution, 34

Geometric Series, 33

Idempotency Laws, 3

Indefinite Integral, 36

indefinite integration, 35

Independent, 7, 10

independent, 6

indeterminate form 0/0, 38

indeterminate form /, 38

integration, 35

Integration by Parts, 37

Integration by Substitution, 36

Intersection, 3

join, 3

L'Hospital's Rule, 38

limits on the iterated integral, 39

Maclaurin Series, 34

meet, 3

moment generating function, 42

moment generating functions, 42

mutually exclusive, 3, 6, 10

Mutually Independent, 7

null set), 3

Pairwise Independent, 7

partial derivatives, 38

Poisson random variable, 34

probability density functions, 42

product, 3

Random Variables, 33

Reading Probabilistic Statements TC  "Reading Probabilistic Statements" \\l 3 , 5

sample space, 2

second partial derivatives, 38

set, 2

set notations, 4

set operations, 3

Set Theory TC  "Set Theory" \\l 3 , 2

Simple Laws and Some Extensions on Sets TC  "Simple Laws and Some Extensions on Sets" \\l 4 , 3

Some Results Involving Limits TC  "Some Results Involving Limits" \\l 3 , 37

Some Results Involving Multivariate Calculus TC  "Some Results Involving Multivariate Calculus" \\l 3 , 38

Some Special Functions TC  "Some Special Functions" \\l 3 , 40

subset, 3

sum, 3

Table of Derivatives for Some Functions, 34

Tables, 12

The Chain Rule, 34

The Three Elementary Methods of Integration, 36

Theorem on Sets, 3

Trees, 13

uniform random variable, 42

Union, 3

universal set, 2

universe, 2

Venn diagram, 13

Venn diagrams, 3