9 781292 039121

ISBN 978-1-29203-912-1

Fundamentals of Probabilitywith Stochastic Processes

Saeed GhahramaniThird Edition

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184 Chapter 4 Distribution Functions and Discrete Random Variables

4.6 STANDARDIZED RANDOM VARIABLES

Let X be a random variable with mean and standard deviation . The random variableX = (X )/ is called the standardized X. We have that

E(X) = E( 1X

)= 1

E(X)

=

= 0,

Var(X) = Var( 1X

)= 1

2Var(X) =

2

2= 1.

When standardizing a random variable X, we change the origin to and the scale tothe units of standard deviation. The value that is obtained for X is independent of theunits in which X is measured. It is the number of standard deviation units by whichX differs from E(X). For example, let X be a random variable with mean 10 feet andstandard deviation 2 feet. Suppose that in a random observation we obtain X = 16; thenX = (16 10)/2 = 3. This shows that the distance of X from its mean is 3 standarddeviation units regardless of the scale of measurement. That is, if the same quantitiesare measured, say, in inches (12 inches = 1 foot), then we will get the same standardizedvalue:

X = 16 12 10 122 12 = 3.

Standardization is particularly useful if two or more random variables with differentdistributions must be compared. Suppose that, for example, a students grade in a prob-ability test is 72 and that her grade in a history test is 85. At rst glance these gradessuggest that the student is doing much better in the history course than in the probabilitycourse. However, this might not be truethe relative grade of the student in probabilitymight be better than that in history. To illustrate, suppose that the mean and standarddeviation of all grades in the history test are 82 and 7, respectively, while these quantitiesin the probability test are 68 and 4. If we convert the students grades to their standarddeviation units, we nd that her standard scores on the probability and history tests aregiven by (72 68)/4 = 1 and (85 82)/7 = 0.43, respectively. These show that hergrade in probability is 1 and in history is 0.43 standard deviation unit higher than theirrespective averages. Therefore, she is doing relatively better in the probability coursethan in the history course. This comparison is most useful when only the means andstandard deviations of the random variables being studied are known. If the distributionfunctions of these random variables are given, better comparisons might be possible.

We now prove that, for a random variable X, the standardized X, denoted by X, isindependent of the units in which X is measured. To do so, let X1 be the observed valueof X when a different scale of measurement is used. Then for some > 0, we have that

184

Chapter 4 Review Problems 185

X1 = X + , and

X1 =X1 E(X1)

X1= (X + )

[E(X) + ]

X+

= [X E(X)]

X= X E(X)

X= X.

EXERCISES

1. Mr. Norton owns two appliance stores. In store 1 the number of TV sets sold bya salesperson is, on average, 13 per week with a standard deviation of ve. Instore 2 the number of TV sets sold by a salesperson is, on average, seven with astandard deviation of four. Mr. Norton has a position open for a person to sell TVsets. There are two applicants. Mr. Norton asked one of them to work in store 1and the other in store 2, each for one week. The salesperson in store 1 sold 10sets, and the salesperson in store 2 sold six sets. Based on this information, whichperson should Mr. Norton hire?

2. The mean and standard deviation in midterm tests of a probability course are 72and 12, respectively. These quantities for nal tests are 68 and 15. What nalgrade is comparable to Velmas 82 in the midterm.

REVIEW PROBLEMS

1. An urn contains 10 chips numbered from 0 to 9. Two chips are drawn at randomand without replacement. What is the probability mass function of their total?

2. A word is selected at random from the following poem of Persian poet and mathe-matician Omar Khayyam (10481131), translated by English poet Edward Fitzger-ald (18081883). Find the expected value of the length of the word.

The moving nger writes and, having writ,Moves on; nor all your Piety nor WitShall lure it back to cancel half a line,Nor all your tears wash out a word of it.

185

186 Chapter 4 Distribution Functions and Discrete Random Variables

3. A statistical survey shows that only 2% of secretaries know how to use the highlysophisticated word processor language TEX. If a certain mathematics departmentprefers to hire a secretary who knows TEX, what is the least number of applicantsthat should be interviewed so as to have at least a 50% chance of nding one suchsecretary?

4. An electronic system fails if both of its components fail. Let X be the time (inhours) until the system fails. Experience has shown that

P(X > t) =(

1 + t200

)et/200, t 0.

What is the probability that the system lasts at least 200 but not more than 300hours?

5. A professor has prepared 30 exams of which 8 are difcult, 12 are reasonable, and10 are easy. The exams are mixed up, and the professor selects four of them atrandom to give to four sections of the course he is teaching. How many sectionswould be expected to get a difcult test?

6. The annual amount of rainfall (in centimeters) in a certain area is a random variablewith the distribution function

F(x) ={

0 x < 5

1 (5/x2) x 5.What is the probability that next year it will rain (a) at least 6 centimeters; (b) atmost 9 centimeters; (c) at least 2 and at most 7 centimeters?

7. Let X be the amount (in uid ounces) of soft drink in a randomly chosen bottlefrom company A, and Y be the amount of soft drink in a randomly chosen bottlefrom company B. A study has shown that the probability distributions of X andY are as follows:

x 15.85 15.9 16 16.1 16.2

P(X = x) 0.15 0.21 0.35 0.15 0.14

P(Y = x) 0.14 0.05 0.64 0.08 0.09

Find E(X), E(Y ), Var(X), and Var(Y ) and interpret them.

8. The fasting blood-glucose levels of 30 children are as follows.

58 62 80 58 64 76 80 80 80 5862 64 76 76 58 64 62 80 58 5880 64 58 62 76 62 64 80 62 76

186

Chapter 4 Review Problems 187

Let X be the fasting blood-glucose level of a child chosen randomly from thisgroup. Find the distribution function of X.

9. Experience shows that X, the number of customers entering a post ofce, duringany period of length t , is a random variable the probability mass function of whichis of the form

p(i) = k (2t)i

i! , i = 0, 1, 2, . . . .

(a) Determine the value of k.(b) Compute P(X < 4) and P(X > 1).

10. From the set of families with three children a family is selected at random, andthe number of its boys is denoted by the random variable X. Find the probabilitymass function and the probability distribution functions of X. Assume that in athree-child family all gender distributions are equally probable.

The following exercise, a truly challenging one, is an example of a game in whichdespite a low probability of winning, the expected length of the play is high.

11. (The Clock Solitaire) An ordinary deck of 52 cards is well shufed and dealtface down into 13 equal piles. The rst 12 piles are arranged in a circle like thenumbers on the face of a clock. The 13th pile is placed at the center of the circle.Play begins by turning over the bottom card in the center pile. If this card is a king,it is placed face up on the top of the center pile, and a new card is drawn from thebottom of this pile. If the card drawn is not a king, then (counting the jack as 11and the queen as 12) it is placed face up on the pile located in the hour positioncorresponding to the number of the card. Whichever pile the card drawn is placedon, a new card is drawn from the bottom of that pile. This card is placed face up onthe pile indicated (either the hour position or the center depending on whether thecard is or is not a king) and the play is repeated. The game ends when the 4th kingis placed on the center pile. If that occurs on the last remaining card, the playerwins. The number of cards turned over until the 4th king appears determines thelength of the game. Therefore, the player wins if the length of the game is 52.

(a) Find p(j), the probability that the length of the game is j . That is, the 4thking will appear on the j th card.

(b) Find the probability that the player wins.(c) Find the expected length of the game.

187

Chapter 5

Special Discrete

Distributions

In this chapter we study some examples of discrete random variables. These ran-dom variables appear frequently in theory and applications of probability, statistics, andbranches of science and engineering.

5.1 BERNOULLI AND BINOMIAL RANDOM VARIABLES

Bernoulli trials, named after the Swiss mathematician James Bernoulli, are perhaps thesimplest type of random variable. They have only two possible outcomes. One outcomeis usually called a success, denoted by s. The other outcome is called a failure, denotedby f . The experiment of ipping a coin is a Bernoulli trial. Its only outcomes are headsand tails. If we are interested in heads, we may call it a success; tails is then a failure.The experiment of tossing a die is a Bernoulli trial if, for example, we are interested inknowing whether the outcome is odd or even. An even outcome may be called a success,and hence an odd outcome a failure, or vice versa. If a fuse is inspected, it is eitherdefective or it is good. So the experiment of inspecting fuses is a Bernoulli trial. Agood fuse may be called a success, a defective fuse a failure.

The sample space of a Bernoulli trial contains two points, s and f . The randomvariable dened by X(s) = 1 and X(f ) = 0 is called a Bernoulli random variable.Therefore, a Bernoulli random variable takes on the value 1 when the outcome of theBernoulli trial is a success and 0 when it is a failure. If p is the probability of a success,then 1 p (sometimes denoted q) is the probability of a failure. Hence the probabilitymass function of X is

p(x) =

1 p q if x = 0p if x = 10 otherwise.

(5.1)

From Chapter 5 of Fundamentals of Probability, with Stochastic Processes, Third Edition. Saeed Ghahramani. Copyright 2005 by Pearson Education, Inc. All rights reserved.

188