# FREUNDS MATHEMATICAL STATISTICS PDF

experiments, experimental psychology, and so forth. Those wishing to participate. From Chapter 1 of John E. Freund's Mathematical Statistics with Applications. John E. Freund's Mathematical Statistics with Applications, Eighth Edition, provides a calculus-based introduction to the theory and application. John E. Freund's Mathematical Statistics With Applications 7th Ed - Ebook download as PDF File .pdf) or read book online. John E. Freund's Mathematical .

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John-E-Freund-s-Mathematical-Statistics-With-Applications-Pearson cepcosojurre.cf - Ebook download as PDF File .pdf), Text File .txt) or. Mathematical Statistics with Applications, 7th Edition Solution Manual for John E. Freund's Mathematical Statistics with Applications 8/e Miller. SIXTH EDITION. John E. Freund's. Mathematical Statistics. IRWIN MILLER. MARYLEES MILLER. Prentice Hall International, Inc.

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No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Freund's Mathematical Statistics with Applications 2. Freund's Mathematical Statistics with Applications, Eighth Edition, provides a calculus- based introduction to the theory and application of statistics, based on comprehensive coverage that reflects the latest in statistical thinking, the teaching of statistics, and current practices.

Freund's Mathematical Statistics with Applications, click button download in the last page 5. You just clipped your first slide! Among the various probability concepts. The results one obtains from an experi- ment. Each outcome in a sample space is called an element of the sample space. We cannot guarantee what will happen on any particular occasion—the car may start and then it may not—but if we kept records over a long period of time.

If a sample space has a finite number of elements. The set of all possible outcomes of an experiment is called the sample space and it is usually denoted by the letter S. The approach to probability that we shall use in this chapter is the axiomatic approach.

It is customary in statistics to refer to any process of observation or measure- ment as an experiment. Sample spaces with a large or infinite number of elements are best described by a statement or rule.

If an experiment consists of one roll of a die and we are interested in which face is turned up. Probability predicts that there is a 30 percent chance for rain that is. In this sense. The different colors are used to emphasize that the dice are distinct from one another. The outcomes of some experiments are neither finite nor countably infinite. Such is the case. If a sample space contains a finite number of elements or an infinite though countable number of elements.

If we assume that distance is a variable that can be measured to any desired degree of accuracy. But even here the number of elements can be matched one-to-one with the whole numbers. A second sample space. Sample spaces are usually classified according to the number of elements that they contain. In the preceding example the sample spaces S1 and S2 contained a finite number of elements.

Probability not be discrete. Rolling a total of 7 with a pair of dice. Solution Among 1. If a sample space consists of a continuum. Continuous sample spaces arise in practice whenever the outcomes of experiments are measure- ments of physical properties.

Solution Among the 36 possibilities. Solution If we let 0 and 1 represent a miss and a hit. Such a subset consists of all the elements of the sample space for which the event occurs. EXAMPLE 6 Construct a sample space for the length of the useful life of a certain electronic component and indicate the subset that represents the event F that the component fails before the end of the sixth year.

Sample space for Example 5. An event is a subset of a sample space. Probability In the same way. This is discussed fur- ther in some of the more advanced texts listed among the references at the end of this chapter. Sample spaces and events. Figure 3. According to our definition. In many problems of probability we are interested in events that are actually combinations of two or more events.

For dis- crete sample spaces. Although the reader must surely be familiar with these terms. When we are dealing with three events. Some of the rules that control the formation of unions. Venn diagrams.

Diagrams showing special relationships among events. Two events having no elements in com- mon are said to be mutually exclusive. The diagram on the right serves to indicate that A is contained in B.

To indicate special relationships among events. Use Venn diagrams to verify the two De Morgan laws: Probability Figure 4. Use Venn diagrams to verify that 3. When A and B are mutually exclusive. Venn diagram. Figure 5. If one event occurs. Let us illustrate this in connection with the frequency interpretation. Before we study some of the immediate consequences of the postulates of prob- ability. Since proportions are always positive or zero.

As far as the frequency interpretation is concerned. Explain why the following assignments of probabilities are not permissible: The following postulates of probability apply only to discrete sample spaces. Probability 4 The Probability of an Event To formulate the postulates of probability.

Taking the third postulate in the simplest case. The second postulate states indirectly that certainty is identified with a probability of 1. If A is an event in a discrete sample space S. How this is done in some special situations is illustrated by the following examples. Proof Let O Since we assume that the coin is balanced.

This is fortunate. To use this theorem. Instead of listing the probabilities of all possible subsets. To assign a probability measure to a sample space. Probability Of course. Letting A denote the event that we will get at least one head. Solution Since the probabilities are all positive. Find P G. The probability measure of Example 10 would be appropriate.

Since there are 13 ways of selecting the face value for the three of a kind and for each of these there are 12 ways of selecting the face value for the pair. If an experiment is such that we can assume equal probabilities for all the sample points.

Probability Here again we made use of the formula for the sum of the terms of an infinite geo- metric progression. EXAMPLE 11 A five-card poker hand dealt from a deck of 52 playing cards is said to be a full house if it consists of three of a kind and a pair.

Solution The number of ways in which we can be dealt a particular full house. If all the five-card hands are equally likely. If A is the union of n of these mutually exclusive N outcomes. If an experiment can result in any one of N different equally likely outcomes. ON represent the individual outcomes in S. Among them. In connection with the frequency interpretation. In practice. Probability would not happen in a million years.

If A and B are any two events in a sample space S. Proof Since A B. For any two events A and B. Probability A B b a c Figure 6. Venn diagram for proof of Theorem 7.

What is the probability that a truck stopped at this roadblock will have faulty brakes as well as badly worn tires? Solution If B is the event that a truck stopped at the roadblock will have faulty brakes and T is the event that it will have badly worn tires.

What is the probability that a family owns either or both kinds of sets? Solution If A is the event that a family in this metropolitan area owns a color television set and B is the event that it owns a HDTV set. What is the probability that a person visiting his dentist will have at least one of these things done to him?

Solution If C is the event that the person will have his teeth cleaned. F is the event that he will have a cavity filled. Prove by induction that Referring to Figure 6. Probability Exercises 5. Use the Venn diagram of Figure 7 with the prob. Use parts a and b of Exercise 3 to show that With reference to the Venn diagram of Figure 7. Use the formula of Theorem 7 to show that a.

C b Postulate 2. Use the Venn diagram of Figure 7 and the method by are A to B that an event will occur. The odds that an event will occur are given by the Venn diagram for Exercises Odds are usually quoted in terms of positive integers having no common factor. Figure 7. Subjective probabilities may be determined by expos- A B ing persons to risk-taking situations and finding the odds at which they would consider it fair to bet on the outcome. Show that if the odds The odds are then converted into probabilities by means b g e of the formula of Exercise See also Exercise Show that if subjective probabilities are determined in this way.

Since the choice of the sample space that is. It is also preferable when we want to refer to several sample spaces in the same example. One of them might apply to all those who are engaged in the private practice of law. G is the event that a person is a law school graduate. Good service Poor service under warranty under warranty In business 10 years or more 16 4 In business less than 10 years 10 20 If a person randomly selects one of these new-car dealers.

L is the event that a person is licensed to practice law. Some ideas connected with conditional probabilities are illustrated in the fol- lowing example. If we let G denote the selection of a dealer who provides good service under warranty.

EXAMPLE 15 A consumer research organization has studied the services under warranty provided by the 50 new-car dealers in a certain city. Probability 6 Conditional Probability Difficulties can easily arise when probabilities are quoted without specification of the sample space.

Generalizing from the preceding. P G T is considerably higher than P G. Of these. Probability For the second question.

This answers the second question and. What is the probability that such an order will be delivered on time given that it was ready for shipment on time? Solution If we let R stand for the event that an order is ready for shipment on time and D be the event that it is delivered on time. Since the probabilities of rolling a 1. Solution If A is the event that the number of points rolled is greater than 3 and B is the event that it is a perfect square.

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Probability Although we introduced the formula for P B A by means of an example in which the possibilities were all equally likely. Solution a If the first card is not replaced before the second card is drawn. If we multiply the expressions on both sides of the formula of Definition 4 by P A. This assumes that we are sampling without replacement.

Probability Thus. EXAMPLE 20 Find the probabilities of randomly drawing two aces in succession from an ordinary deck of 52 playing cards if we sample a without replacement. Note that P R D. If 3 of the fuses are selected at random and removed from the box in succession without replacement.

Solution If A is the event that the first fuse is defective. Theorem 9 can easily be generalized so that it applies to more than two events. B is the event that the second fuse is defective. Probability b If the first card is replaced before the second card is drawn. If A is the event that a head occurs on each of the first two tosses. B is the event that a tail occurs on the third toss.

Probability 7 Independent Events Informally speaking. If two events are not independent. For mathematical convenience. In the deriva- tion of the formula of Definition 5. To extend the concept of independence to more than two events. In connection with Definition 5.

If A and B are independent. For three events A. In Exercises 22 and 23 the reader will be asked to show that if A and B are independent. Events A1. Venn diagram for Example A and C are independent.

Verify that A and B are independent. Solution As can be seen from the diagram. The following is a simple example in which there is one intermediate stage consisting of two alternatives: Thus we can multiply. Inasmuch as the tosses are independent. Of course.

Making use of the formula of part a of Exercise 3. What is the probability that the construction job will be completed on time? Solution If A is the event that the construction job will be completed on time and B is the event that there will be a strike. The prob- abilities are 0. Solution 1 a The probability of a head on each toss is and the three outcomes are inde- 2 pendent. Probability Then.

If the events B1. It requires the following theorem. If 9 percent of the cars from agency 1 need an oil change. With reference to the preceding example. Solution If A is the event that the car needs an oil change.

To answer ques- tions of this kind. Substi- tuting these values into the formula of Theorem If a rental car delivered to the consulting firm needs an oil change. A formal proof of Theorem 12 consists. In words. If B1. Solution Substituting the probabilities on the previous page into the formula of Theorem Given three events A. Solution Let D and p represent the events that a person randomly selected from the given population.

Refer to Figure 10 to show that if A is independent b need not be equal to 1. Show that the postulates of probability are satisfied Substituting the given probabilities into the formula of Theorem A test has been developed that will be positive.

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Show that if events A and B are independent. Find the probability that a person tested as positive does not have the disease. Exercises If events A. Duplicating the method of proof of Theorem B and A is independent of C. Show that if events A and B are dependent. If A1. Diagram for Exercises Probabilities were first considered in games of chance. The entire theory of statistics is based on the notion of probability.

Prove that the probability of at least one match is given by Figure Motivated by problems associated with games of chance. For any event A. The assumption of equal likelihood fails when we attempt. Together with the rules given in Section 5. Probability Incorporated www. An are independent events.

Some of them went as far as to postulate some of these rules entirely on the basis of experience. But differ- ences arose among gamblers about probabilities. Reprinted with permission. It seems remarkable that the entire structure of probability and. With this motivation. The postulates of probability given in Section 4 satisfy this criterion. Prove Theorem 12 by making use of the following 0.

The die 0. Assume that the probability of it coming up on the side numbered i is the same for each value of i. Proof The proof follows immediately by iterating in Definition 5. Under this assumption. For exam- ple. As another example. A more recently employed method of calculating probabilities is called the sub- jective method. This idea gives rise to the fre- quency interpretation of probabilities. A series system is one in which all com- ponents are so interrelated that the entire system will fail if any one or more of its components fails.

The reliability of a product is the probability that it will function within specified limits for a specified period of time under specified environmental conditions. Application of the frequency interpretation requires a well-documented his- tory of the outcomes of an event over a large number of experimental trials. We shall assume that the components connected in a series system are indepen- dent.

In the absence of such a history. An important application of probability theory relates to the theory of reliabil- ity. The reliability of a component or system can be defined as follows. A parallel system will fail only if all its components fail. Find the reliability of the system. Again applying Definition 5. Combination of series and parallel systems.

If the system complexity were increased so it now has 10 such components. E can be replaced by an equivalent component. Solution The parallel subsystem C. Probability Theorem 14 vividly demonstrates the effect of increased complexity on reliability. One way to improve the reliability of a series system is to introduce parallel redundancy by replacing some or all of its components by several components con- nected in parallel. If a system consists of n independent components connected in parallel.

A coin is tossed once. Santa Mon- ica. If the larger subsets of the sample space. Car 4 is three years old correspond to each of the following events: V is the event that it costs An electronics firm plans to build a research labora.

Using the notation in which H. Car 5 is new and has no air-conditioning. Car 1 is new and has air-conditioning. Among the eight cars that a dealer has in his show. If a customer twice more. Let A be the event b the union of the sets of parts b and c. San Diego. B repre.

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Car 2 is one year old and has air. Car 7 is two years old and has no air- conditioning. Santa Barbara.

U is the event that it has a fireplace. A resort hotel has two station wagons. An electronic game contains three components F stands for the event ing. G stands for the event that each carries the same num- Car 3 is two years old and has air-conditioning and power ber of passengers.

If Ms.

## John E. Freund's Mathematical Statistics with Applications Solutions Manual

Brown downloads one of the houses advertised for sale in a Seattle newspaper on a given Sunday. Draw a suitable Venn diagram and fill in the numbers associated with the various regions. Express in words the events represented by Figure In Figure With reference to Exercise 48 and Figure A and C.

L is the event that a driver has liability c transmission repairs or new tires. Among visitors to Disneyland. Diagram for Exercise With reference to Exercise 46 and Figure In a group of college students. A and B. Express in words what events are represented by d new tires.

Using a Venn diagram and filling in the number of shoppers associated with the 2 1 3 various regions. How many of these Figure Venn diagram for Exercise Describe the sample space and determine 4 3 a how many elements of the sample space correspond to the event that the 3 appears on the kth roll of the die.

A market research organization claims that. An experiment consists of rolling a die until a 3 1 appears. Express symbolically the sample space S that consists Figure Using the notation d regions 1 and 4 together. Probability c regions 1. Goodyear tires. Venn diagram with three circles like that of Figure 4 and filling in the numbers associated with the various regions.

Matterhorn ride. Two cards are randomly drawn from a deck of 52 tie the game is 0. Find the probabili- ity of the machine will be rated ties of getting Find the probability that both cards will be tie the game is 0. General tires. An experiment has five possible outcomes. Four candidates are seeking a vacancy on a school 0. Goodrich tires. Drawing a d average or better. A hat contains 20 white slips of paper numbered from and E. If A is twice as likely to be elected as B.

Check whether the fol. Find the a stayed for at least 3 hours. If A and B are mutually exclusive. The probabilities that the serviceability of a new b A will not win? X-ray machine will be rated very difficult. Explain why there must be a mistake in each of the will download following statements: Find the probabilities that the serviceabil.

Michelin tires. Suppose that each of the 30 points of the sample space find how many of the visitors to Disneyland 1 of Exercise 40 is assigned the probability In a poker game. Find the probabilities that it It is also known that 59 of the stu- b four of a kind four cards of equal face value. The probability that the older What is the probability that it will divide the A line segment of length l is divided by a point bility that a person visiting Disneyland will go on at least selected at random within the segment.

Probability a two pairs any two distinct face values occurring If one of these ability that he will go on the Matterhorn ride is 0. Explain on the basis of the various rules of of the two assistants will be absent on any given day is Exercises 5 through 9 why there is a mistake in each of 0. Find the probabilities of getting What is the probability that a student selected at ran- a two pairs.

Use the formula of Exercise Matterhorn ride is 0. In a game of Yahtzee. A right triangle has the legs 3 and 4 units. What is the probability that at least one of Amsterdam is 0. Among the 78 doctors on the staff of a hospital. A biology professor has two graduate assistants help- d four of a kind. Suppose that if a person travels to Europe for the segment exactly in half? Suppose that if a person visits Disneyland. What is the prob. What is the proba- At Roanoke College it is known that 13 of the stu- exactly twice.

Find the bility that it will rain or snow is 0. For married couples living in a certain suburb. Using the method of Exercise Some of them are college graduates and some are not. Discuss the consistency of the corresponding To be consistent see Exercise There are two Porsches in a road race in Italy.

If the order in which the applicants are interviewed by b Verify that the sum of these probabilities is 1. Are the graduates graduates corresponding probabilities consistent? Estimate the prob- ability that A: A bin contains balls. Use the formula of Exercise 15 to convert each of the There are 90 applicants for a job with the news depart- subjective probabilities.

Use the results of Exercise 76 to verify that chosen integer will have a value less than 1. If subjective probabilities are determined by the convert each of the following probabilities to odds: G is the event that the If two balls are selected from the the odds are 5 to 1 that we will not get a meaningful word bin without replacement.

If the coin is tossed d go on the Matterhorn ride and the Jungle Cruise given three times. What is the probability of surviving both dealer has in stock. Note that the region repre- a a hit followed by two misses.

If a person randomly picks 4 of the 15 gold coins a a month is 0. It is felt that the probabilities are 0. The probability of surviving a certain transplant oper. With reference to Figure Crates of eggs are inspected for blood clots by ran- tion. Medical records show that one out of 10 persons in that a crate will be shipped if it contains eggs.

If all three eggs are good. A shipment of 1. Find the probabilities that a rainy fall day is she will also pay promptly the next month. If A is the event that and 0. A balanced die is tossed twice. If 12 persons in 10 have blood clots? Assume that the probability of paying or not c two rainy days and then two sunny days. Draw a Venn diagram and fill in the probabilities b two tails and a head in that order?

Find the probability that a ride the Monorail given that he will go on the Jun. A department store that bills its charge-account cus- a rainy fall day is followed by a rainy day is 0. What is the probability If a patient survives the operation. In part c use the formula of Exercise Use the formula of Exercise 19 to find the probability b What is the probability that a customer who does not of randomly choosing without replacement four healthy pay promptly one month will also not pay promptly the guinea pigs from a cage containing 20 guinea pigs.

B is the event T. A sharpshooter hits a target with probability 0. Suppose that in Vancouver. Assuming independence. At an electronics plant, it is known from past expe- the probability that a one-car accident is incorrectly rience that the probability is 0. If 70 percent of all new actually due to faulty brakes? With reference to Example 25, suppose that we dis- cover later that the job was completed on time.

What is It is known from experience that in a certain indus- the probability that there had been a strike? In a certain community, 8 percent of all adults over percent are over fringe issues. Also, 45 percent of the 50 have diabetes. If a health service in this community disputes over wages are resolved without strikes, 70 per- correctly diagnoses 95 percent of all persons with dia- cent of the disputes over working conditions are resolved betes as having the disease and incorrectly diagnoses 2 without strikes, and 40 percent of the disputes over fringe percent of all persons without diabetes as having the dis- issues are resolved without strikes.

What is the probabil- ease, find the probabilities that ity that a labor—management dispute in this industry will a the community health service will diagnose an adult be resolved without a strike? In a T-maze, a rat is given food if it turns left and an having diabetes actually has the disease. On the first trial there is a 50—50 chance that a rat will turn either way; then, if it An explosion at a construction site could have receives food on the first trial, the probability is 0.

Interviews with on the first trial, the probability is 0.

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What is the probability that a rat will to the estimates that such an explosion would occur with turn left on the second trial? With reference to Exercise , what is the probabil- carelessness, and 0. It is also ity that if a labor—management dispute in this industry is felt that the prior probabilities of the four causes of the resolved without a strike, it was over wages?

Based on all this The probability that a one-car accident is due to information, what is faulty brakes is 0.

## Mathematics

U makes mate of P Y to estimate P M. A series system consists of three components, each in an order three times in a hundred. If U, V, and W fill, having the reliability 0.

Find the reliability of the the probabilities that system. Find the reliability of a series systems having five by U; components with reliabilities 0. What must be the reliability of each component in a An art dealer receives a shipment of five old paint- series system consisting of six components that must have ings from abroad, and, on the basis of past experience, a system reliability of 0.

Referring to Exercise , suppose now that there 0. Since the cost of authentication is fairly 0. If it turns out Suppose a system consists of four components, con- that this painting is a forgery, what probability should she nected in parallel, having the reliabilities 0.

Find the system reliability. Referring to Exercise , suppose now that the sys- To get answers to sensitive questions, we sometimes tem has five components with reliabilities 0. Suppose, for instance, that we want to determine what percentage of the students at a large university smoke A system consists of two components having the reli- marijuana.

Then, we let each stu- components, each having the reliability 0. Find the sys- dent in the sample interviewed select one of the 20 cards tem reliability. A series system consists of two components having a Establish a relationship between P Y , the probabil- the reliabilities 0. References Among the numerous textbooks on probability theory Freund, J. Mineola, Applications, Vol. I, 3rd ed. Dover Publications, Inc. San Francisco: Holden Day, Inc. Modeling , Uncertainty. Reading, Mass.: Addison-Wesley Publish- Nosal, M.

Philadel- ing Company, Inc. Saunders Company, Draper, N. An Introductory Course. Markam Publishing More advanced treatments are given in many texts, Company, , for instance,. Hoel, P. Upper Probability Theory. Prentice Hall, Pacific Palisades, Calif.: Goodyear Publishing Johnson, R.

Upper Saddle River, N. Parzen, E. Macmillan Publishing Company, For instance, when we roll a pair of dice we are usually interested only in the total, and not in the outcome for each die; when we interview a randomly chosen married couple we may be interested in the size of their family and in their joint income, but not in the number of years they have been married or their total assets; and when we sample mass-produced light bulbs we may be interested in their durability or their brightness, but not in their price.

In each of these examples we are interested in numbers that are associated with the outcomes of chance experiments, that is, in the values taken on by random vari- ables. In the language of probability and statistics, the total we roll with a pair of dice is a random variable, the size of the family of a randomly chosen married couple and their joint income are random variables, and so are the durability and the brightness of a light bulb randomly picked for inspection.

To be more explicit, consider Figure 1, which pictures the sample space for an experiment in which we roll a pair of dice, and let us assume that each of the 36 1 possible outcomes has the probability Note, however, that in Figure 1 we have attached a number to each point: For instance, we attached the number 2 to the point 1, 1 , the number 6 to the point 1, 5 , the number 8 to the point 6, 2 , the number 11 to the point 5, 6 , and so forth.

Evidently, we associated with each point the value of a random variable, that is, the corresponding total rolled with the pair of dice.

From Chapter 3 of John E. If S is a sample space with a probability measure and X is a real-valued function defined over the elements of S, then X is called a random variable. In this chapter we shall denote random variables by capital letters and their values by the corresponding lowercase letters; for instance, we shall write x to denote a value of the random variable X.

EXAMPLE 1 Two socks are selected at random and removed in succession from a drawer contain- ing five brown socks and three green socks. List the elements of the sample space, the corresponding probabilities, and the corresponding values w of the random variable W, where W is the number of brown socks selected.

List the elements of the sample space that are presumed to be equally likely, as this is what we mean by a coin being balanced, and the corresponding values x of the random variable X, the total number of heads. Solution If H and T stand for heads and tails, the results are as shown in the following table:. The fact that Definition 1 is limited to real-valued functions does not impose any restrictions. If the numbers we want to assign to the outcomes of an experiment are complex numbers, we can always look upon the real and the imaginary parts sepa- rately as values taken on by two random variables.

In all of the examples of this section we have limited our discussion to discrete sample spaces, and hence to discrete random variables, namely, random variables whose range is finite or countably infinite.

Continuous random variables defined over continuous sample spaces will be taken up in Section 3. The probabilities associated with all possible values of X are shown in the follow- ing table:. For instance, for the total rolled with a pair of dice we could write.

A function can serve as the probability distribution of a dis- crete random variable X if and only if its values, f x , satisfy the conditions 1. Since these values are all nonnegative, the first condition 6 7. Of course, whether any given random variable actually has this probability distribution is an entirely different matter. In some problems it is desirable to present probability distributions graphi- cally, and two kinds of graphical presentations used for this purpose are shown in Figures 2 and 3.

The one shown in Figure 2, called a probability histogram, repre- sents the probability distribution of Example 3. The height of each rectangle equals. Probability histogram. Bar chart. Probability Distributions and Probability Densities f x 6 16 4 16 1 16 x 0 1 2 3 4 Number of heads Figure 2. Since each rectangle of the probability histogram of Figure 2 has unit width.

This can be done even when the rectangles of a probability histogram do not all have unit width by adjusting the heights of the rectangles or by modifying the vertical scale.

As in Figure 2. There are many problems in which it is of interest to know the probability that the value of a random variable is less than or equal to some real number x. In this chapter. The graph of Figure 3 is called a bar chart. Probability Distributions and Probability Densities corresponding probabilities.

If X is a discrete random variable. There are certain advantages to identifying the areas of the rectangles with the probabilities.

If we are given the probability distribution of a discrete random variable. The values F x of the distribution function of a discrete ran- dom variable X satisfy the conditions 1. Probability Distributions and Probability Densities Based on the postulates of probability and some of their immediate consequences. Note that at all points of discontinuity the distribution function takes on the greater of the two values. Solution Based on the probabilities given in the table in Section 1.

Graph of the distribution function of Example 6. In the remainder of this chapter we will be concerned with continuous ran- dom variables and their distributions and with problems relating to the simultaneous occurrence of the values of two or more random variables.

Solution Making use of Theorem 3.

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To this end. Probability Distributions and Probability Densities We can also reverse the process illustrated in the two preceding examples. For each of the following. Construct a probability histogram for each of the fol- given values can serve as the values of a probability dis.

Prove Theorem 2. If X has the distribution function tion can serve as the probability distribution of a random variable with the given range: Find the distribution function of the random variable with the given range. Probability Distributions and Probability Densities Exercises 1.

Show that there are no values of c such that To illustrate. Probability Distributions and Probability Densities If X has the distribution function When the value of a ran- dom variable is given directly by a measurement or observation.

The problem of defining probabilities in connection with continuous sample spaces and continuous random variables involves some complications. EXAMPLE 8 Suppose that we are concerned with the possibility that an accident will occur on a freeway that is kilometers long and that we are interested in the probability that it will occur at a given location. With reference to Example 4. The outcomes of experiments are represented by the points on line segments or lines.Freund's Mathematical Statistics with Applications homework has never been easier than with Chegg Study.

Car 7 is two years old and has no air- conditioning. More recently, the work of R. Addison-Wesley Publishing Company. Venn diagram for proof of Theorem 7.

Use this generalized definition of binomial coefficients by expressing all these multinomial coefficients in terms to evaluate of factorials and simplifying algebraically. An experiment consists of rolling a die until a 3 1 appears.

Solution Using Theorem 6. Prentice Hall,