Applied Ordinal Logistic Regression Using Stata 1st Edition Xing Liu - Solutions

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4.20. Suppose that X is uniformly distributed over [-a, +a], where a > 0. When- ever possible, determine a so that the following are satisfied. (a) P(X > 1) = (d) P(X < 0.3 (b) P(X> 1)(c) P(X 1).
4.19. Let X be a binomially distributed random variable based on 10 repetitions of an experiment. If p = 0.3, evaluate the following probabilities using the table of the binomial distribution in the Appendix. (a) P(X 8) (b) P(X = 7) (c) P(X > 6).
4.18. Let X be the life length of an electronic device (measured in hours). Suppose that X is a continuous random variable with pdf f(x) = k/x", 2000 x 10,000. (a) For n = 2, determine k. (b) For n=3, determine k. (c) For general n, determine k. (d) What is the probability that the device will fail
4.17. Each of the following functions represents the cdf of a continuous random variable. In each case F(x) = 0 for xb, where [a, b] is the indicated interval. In each case, sketch the function F, determine the pdf fand sketch it. Also verify that fis a pdf. (a) F(x) = x/5, 0 < x < 5 (c) F(x) (b)
4.16. The diameter on an electric cable, say X, is assumed to be a continuous random variable with pdf f(x) = 6x(1-x), 0 x 1.(a) Check that the above is a pdf and sketch it. (b) Obtain an expression for the cdf of X and sketch it. (c) Determine a number b such that P(X b). (d) Compute P(X < <
4.15. Let X be a continuous random variable with pdf / given by: f(x)-ax, -a 0 x 1, 1 x 2, =-ax + 3a, 2 x 3, - 0, elsewhere. (a) Determine the constanta. (b) Determine F, the cdf, and sketch its graph. (c) If X1, X2, and X3 are three independent observations from X, what is the prob- ability that
4.14. The percentage of alcohol (100X) in a certain compound may be considered as a random variable, where X, 0 < x < 1, has the following pdf: f(x) = 20x (1-x), 0 < x < 1. (a) Obtain an expression for the cdf F and sketch its graph. (b) Evaluate P(X }). (c) Suppose that the selling price of the
4.13. Suppose that the graph in Fig. 4.16 represents the pdf of a random variable X. (a) What is the relationship between a and b? (b) If a > 0 and b > 0, what can you say about the largest value which b may assume? (See Fig. 4.16.) f(x) (a,b) x--a x=b FIGURE 4.16
4.12. Suppose that fand g are pdf's on the same interval, say a xb. (a) Show that+g is not a pdf on that interval. (b) Show that for every number 3,0
4.11. The continuous random variable X has pdf f(x) 3x2, -1 x 0. If -1x0. b is a number satisfying -1 b|X < b/2).
4.10. Let X be the life length of an electron tube and suppose that X may be repre- sented as a continuous random variable with pdf f(x) bebe, x0. Let p; = PUX.
4.9. The continuous random variable X has pdf f(x) = x/2, 0 x 2. Two independent determinations of X are made. What is the probability that both these determinations will be greater than one? If three independent determinations had been made, what is the probability that exactly two of these are
4.8. (Properties of the binomial probabilities.) In the discussion of Example 4.8 a general pattern for the binomial probabilities () p* (1 - p) was suggested. Let us denote these probabilities by p.(k). (a) Show that for 0 < k < n we have P.(k+1)/p.(k) = [(n - k)/(k + 1)] [p/(1 [(n-k)/(k - p)].
4.7. Evaluate P(X-5), where X is the random variable defined in Example 4.10. Suppose that n10, n2 15, pi 0.3 and p2 = 0.2. -
4.6. Rockets are launched until the first successful launching has taken place. If this does not occur within 5 attempts, the experiment is halted and the equipment in- spected. Suppose that there is a constant probability of 0.8 of having a successful launching and that successive attempts are
4.5. Suppose that twice as many items are produced (per day) by machine 1 as by machine 2. However, about 4 percent of the items from machine 1 tend to be defective while machine 2 produces only about 2 percent defectives. Suppose that the daily output of the two machines is combined. A random
4.4. Consider a random variable X with possible outcomes: 0, 1, 2,... Suppose that P(X j) (1a)a', J 0, 1, 2,... (a) For what values of a is the above model meaningful? (b) Verify that the above does represent a legitimate probability distribution. (c) Show that for any two positive integers s and
4.3. Suppose that the random variable X has possible values 1, 2, 3,..., and P(X =j) = 1/2,j=1, 2,...(a) Compute P(X is even). (b) Compute P(X 5). (c) Compute P(X is divisible by 3).
4.2. From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let X be the number of defectives found. Obtain the probability distribution of X if (a) the items are chosen with replacement, (b) the items are chosen without replacement.
4.1. A coin is known to come up heads three times as often as tails. This coin is tossed three times. Let X be the number of heads that appear. Write out the probability distri- bution of X and also the cdf. Make a sketch of both.
3.39. A printing machine can print n "letters," say a1,...,. It is operated by electrical impulses, each letter being produced by a different impulse. Assume that there exists a constant probability p of printing the correct letter and also assume independence. One of the impulses, chosen at
3.38 Urn I contains a white and 3 black balls while Urn 2 contains 8 white and a black balls. One ball is chosen (from one of the urns) and is then returned to that urn. If the chosen ball is white, choose the next ball from Urn 1; if the chosen ball is black, choose the next one from Urn 2.
3.37. Urn 1, Urn 2,..., Urn n each contain a white and black balls. One ball is taken from Urn 1 into Urn 2 and then one is taken from Urn 2 into Urn 3, etc. Finally, one ball is chosen from Urn n. If the first ball transferred was white, what is the prob-ability that the last ball chosen is white?
3.36 A fair coin is tossed 2n times. (a) Obtain the probability that there will be an equal number of heads and tails. (b) Show that the probability computed in (a) is a decreasing function of n.
3.35. Three newspapers, A, B, and C, are published in a city and a recent survey of readers indicates the following: 20 percent read 4, 16 percent read B, 14 percent read C, 8 percent read A and B, 5 percent read A and C, and 2 percent read A, B, and C. For one adult chosen at random, compute the
3.34. The following (somewhat simple-minded) weather forecasting is used by an amateur forecaster. Each day is classified as "dry" or "wet" and the probability that any given day is the same as the preceding one is assumed to be a constant p (0 < p < 1). Based on past records, it is supposed that
3.33. Four radio signals are emitted successively. If the reception of any one signal is independent of the reception of another and if these probabilities are 0.1, 0.2, 0.3, and 0.4, respectively, compute the probability that k signals will be received for k = 0, 1, 2, 3, 4.
3.32. The probability that a system becomes overloaded is 0.4 during each run of an experiment. Compute the probability that the system will cease functioning in three independent trials of the experiment if the probabilities of failure in 1, 2, or 3 trials equal 0.2, 0.5, and 0.8, respectively.
3.31. An electrical system consists of two switches of type 4, one of type B, and four of type C, connected as in Fig. 3.12. Compute the probability that a break in the circuit cannot be eliminated with key K if the switches A, B, and C are open (i.e., out of order) with probabilities 0.3, 0.4, and
3.30. A vacuum tube may come from any one of three manufacturers with prob- abilities p = 0.25, p2 = 0.50, and ps 0.25. The probabilities that the tube will function properly during a specified period of time equal 0.1, 0.2, and 0.4, respectively. for the three manufacturers. Compute the
3.29. Prove: If P(A|B) > P(A) then P(B | A) > P(B).
3.28. Suppose that a mechanism has N tubes, all of which are needed 'for its func- tioning. To locate a malfunctioning tube one replaces each tube, successively, with a new one. Compute the probability that it will be necessary to check N tubes if the (con- stant) probability is p that a tube is
3.27. Whenever an experiment is performed, the occurrence of a particular event A equals 0.2. The experment is repeated, independently, until A occurs. Compute the probability that it will be necessary to carry out a fourth experiment.
3.26. Finish the analysis of the example given in Section 3.2 by deciding which of the types of candy jar, A or B, is involved, based on the evidence of two pieces of candy which were sampled.
3.25. An electronic assembly consists of two subsystems, say A and B. From previous testing procedures, the following probabilities are assumed to be known: P(A fails) = 0.20, P(B fails alone) - 0.15. P(A and B fail) 0.15, Evaluate the following probabilities. (a) P(A fails | B has failed), (b) P(A
3.24. Show that the multiplication theorem P(B) = P(A|B)P(B), established for two events, may be generalized to three events as follows: P(ABC) = P(A|BNC)P(B| C)P(C).
3.23. If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability.)
3.22. By verifying Eq. (3.2), show that for fixed A, P(B| A) satisfies the various postu- lates for probability.
3.21. Two machines, A, B, being operated independently, may have a number of breakdowns each day. Table 3.2 gives the probability distribution of breakdowns for each machine. Compute the following probabilities.(a) A and B have the same number of breakdowns. (b) The total number of breakdowns is
3.20. In Fig. 3.11(a) and (b), assume that the probability of each relay being closed is p and that each relay is open or closed independently of any other relay. In each case find the probability that current flows from L to R.Number of breakdowns 0 1 2 3 5 A 0.1 0.2 0.3 0.2 0,09 0.07 0.04 B 0.3
3.19. Prove that if A and B are independent events, so are A and B, A and B, A and B. (a) R FIGURE 3.11 (b)
3.18. Verify that the number of conditions listed in Eq. (3.8) is given by 2* - 1.
3.17. It is found that in manufacturing a certain article, defects of one type occur with probability 0.1 and defects of a second type with probability 0.05. (Assume inde- pendence between types of defects.) What is the probability that: (a) an article does not have both kinds of defects? (b) an
3.16. Two dice are rolled. Given that the faces show different numbers, what is the probability that one face is 4?
3.15. Each of two persons tosses three fair coins. What is the probability that they obtain the same number of heads?
3.14. A die is thrown n times. What is the probability that "6" comes up at least once in the throws?
3.13. A binary number is one composed only of the digits zero and one. (For example, 1011, 1100, etc.) These numbers play an important role in the use of electronic computers. Suppose that a binary number is made up of n digits. Suppose that the probability of an incorrect digit appearing is p and
3.12. A die is tossed, and independently, a card is chosen at random from a regular deck. What is the probability that: (a) the die shows an even number and the card is from a red suit? (b) the die shows an even number or the card is from a red suit?
3.11. Three components of a mechanism, say C1, C2, and Cs are placed in series (in a straight line). Suppose that these mechanisms are arranged in a random order. Let R be the event (C is to the right of C. and let S be the event (Cs is to the right of C). Are the events R and S independent? Why?
310. Let A and B be two events associated with an experiment. Suppose that P(A)= 0.4 while P(AUB) 0.7. Let P(B) p. (a) For what choice of p are A and B mutually exclusive? (b) For what choice of p are A and B independent?
3.9. In a bolt factory, machines A, B, and C manufacture 25, 35, and 40 percent of the total output, respectively. Of their outputs, 5, 4, and 2 percent, respectively, are defective bolts. A bolt is chosen at random and found to be defective. What is the prob- ability that the bolt came from
3.8. A bag contains three coins, one of which is coined with two heads while the other two coins are normal and not biased. A coin is chosen at random from the bag and tossed four times in succession. If heads turn up each time, what is the probability that this is the two-headed coin?
3.7. Suppose that we have two urns, 1 and 2, each with two drawers. Urn 1 has a gold coin in one drawer and a silver coin in the other drawer, while urn 2 has a gold coin in each drawer. One urn is chosen at random; then a drawer is chosen at random from the chosen urn. The coin found in this
3.6. Twenty items, 12 of which are defective and 8 nondefective, are inspected one after the other. If these items are chosen at random, what is the probability that: (a) the first two items inspected are defective? (b) the first two items inspected are nondefective? (c) among the first two items
3.5. Suppose that A and B are independent events associated with an experiment. If the probability that A or B occurs equals 0.6, while the probability that A occurs equals 0.4, determine the probability that B occurs.
3.4. In the above problem the tubes are checked by drawing a tube at random, testing it and repeating the process until all 4 bad tubes are located. What is the probability that the fourth bad tube will be located (a) on the fifth test? (b) on the tenth test?
3.3. A box contains 4 bad and 6 good tubes. Two are drawn out together. One of them is tested and found to be good. What is the probability that the other one is also good?
3.2. Two defective tubes get mixed up with two good ones. The tubes are tested, one by one, until both defectives are found. (a) What is the probability that the last defective tube is obtained on the second test? (b) What is the probability that the last defective tube is obtained on the third
3.1. Urn I contains x white and y red balls. Urn 2 contains z white and red balls. A ball is chosen at random from urn 1 and put into urn 2. Then a ball is chosen at random from urn 2. What is the probability that this ball is white?
2.22. numbers (0 < < 10) are chosen at random (without replacement) from the numbers 0, 1, 2,...,9. What is the probability that no two are equal?
2.21. A lot contains n articles. If it is known that of the articles are defective and the articles are inspected in a random order, what is the probability that the kth article (kr) inspected will be the last defective one in the lot?
2.20. A certain chemical substance is made by mixing 5 separate liquids. It is proposed to pour one liquid into a tank, and then to add the other liquids in turn. All possible combinations must be tested to see which gives the best yield. How many tests must be performed?
2.19. From 6 positive and 8 negative numbers, 4 numbers are chosen at random (without replacement) and multiplied. What is the probability that the product is a positive number?
2.18. One integer is chosen at random from the numbers 1, 2,..., 50. What is the probability that the chosen number is divisible by 6 or by 8?
2.17. How many subsets can be formed, containing at least one member, from a set of 100 elements?
2.16. A box contains tags marked 1, 2,..., n. Two tags are chosen at random. Find the probability that the numbers on the tags will be consecutive integers if (a) the tags are chosen without replacement, (b) the tags are chosen with replacement.
2.15. Suppose that )a and (2)b. Express (19) in terms of a andb. [Hint: Do not evaluate the above expressions to solve this problem.]
2.14. From the lettersa, b,c, d,e, and f how many 4-letter code words may be formed if, (a) no letter may be repeated? (b) any letter may be repeated any number of times?
2.13. Suppose that from N objects we choosen at random, with replacement. What is the probability that no object is chosen more than once? (Suppose that n < N.)
2.12. A mechanism may be set at any one of four positions, saya, b,c, andd. There are 8 such mechanisms which are inserted into a system. (a) In how many ways may this system be set? (b) Assume that these mechanisms are installed in some preassigned (linear) order. How many ways of setting the
2.11. There are 12 ways in which a manufactured item can be a minor defective and 10 ways in which it can be a major defective. In how many ways can 1 minor and I major defective occur? 2 minor and 2 major defectives?
2.10. A complex mechanism may fail at 15 stages. If it fails at 3 stages, in how many ways may this happen?
2.9. An inspector visits 6 different machines during the day. In order to prevent operators from knowing when he will inspect he varies the order of his visits. In how many ways may this be done?
2.8. A product is assembled in three stages. At the first stage there are 5 assembly lines, at the second stage there are 4 assembly lines, and at the third stage there are 6 assembly lines. In how many different ways may the product be routed through the assembly process?
2.7. If from the lot of articles described in Problem 2.6 two articles are chosen (with- out replacement), find the probability that: (a) both are good, (c) at least one is good, (e) exactly one is good, (b) both have major defects, (d) at most one is good, (f) neither has major defects, (g)
2.6. A lot consists of 10 good articles, 4 with minor defects, and 2 with major defects. One article is chosen at random. Find the probability that: (a) it has no defects, (b) it has no major defects, (c) it is either good or has major defects.
2.5. Ten chips numbered 1 through 10 are mixed in a bowl. Two chips numbered (X, Y) are drawn from the bowl, successively and without replacement. What is the probability that X+Y 10?
2.4. A shipment of 1500 washers contains 400 defective and 1100 nondefective items. Two-hundred washers are chosen at random (without replacement) and classified. (a) What is the probability that exactly 90 defective items are found? (b) What is the probability that at least 2 defective items are
2.3. (a) Suppose that the three digits 1, 2, and 3 are written down in random order. What is the probability that at least one digit will occupy its proper place? (b) Same as (a) with the digits 1, 2, 3, and 4. (c) Same as (a) with the digits 1, 2, 3,..... [Hint: Use (1.7).] (d) Discuss the answer
2.2. Ten persons in a room are wearing badges marked 1 through 10. Three persons are chosen at random, and asked to leave the room simultaneously. Their badge number is noted. (a) What is the probability that the smallest badge number is 5? (b) What is the probability that the largest badge number
2.1. The following group of persons is in a room: 5 men over 21, 4 men under 21, 6 women over 21, and 3 women under 21. One person is chosen at random. The following events are defined: 4 = {the person is over 21); B = (the person is under 21); C = (the person is male); D = (the person is female).
1.19. A mechanism has two types of parts, say I and II. Suppose that there are two of type I and three of type II. Define the events A, k-1, 2, and B. j 1, 2, 3 as follows: A: the kth unit of type I is functioning properly; B: the jth unit of type II is functioning properly. Finally, let C
1.18. An installation consists of two boilers and one engine. Let the event A be that the engine is in good condition, while the events B, (k-1, 2) are the events that the kth boiler is in good condition. The event C is that the installation can operate. If the installation is operative whenever
1.17. Suppose that A, B, and C are events such that P(A) P(AB) P(CB)-0, and P(AC) least one of the events A, B, or C occurs.
1.16. Suppose that A and B are events for which P(A) = x, P(B) = y, and P(AB). Express each of the following probabilities in terms of x, y, and z. (a) P(AUB) (b) P(AB) (c) P(AUB) (d) P(AB) P(B) = P(C) - Evaluate the probability that at
1.15. A certain type of electric motor fails either by seizure of the bearings, or by burning out of the electric windings, or by wearing out of the brushes. Suppose that seizure is twice as likely as burning out, which is four times as likely as brush wearout. What is the probability that failure
1.14. Theorem 1.3 deals with the probability that at least one of the two events A or Boccurs. The following statement deals with the probability that exactly one of the events A or B occurs. Show that [P(AB) U (B7)] = P(A) + P(B) 2P(A ) B).
1.13. (a) Show that for any two events, 41 and 42, we have P(A1 U A2) P(A1) + P(A2). (b) Show that for any w events A...A.. we have P(A UUA) P(A) + ... + P(A). [Hint: Use mathematical induction. The result stated in (b) is called Boole's inequality.]
1.12. Prove Theorem 1.4.
Present a table containing the estimated logit coefficients and odds ratios for the contextual model.
Use the meglm command with the logit option to fit the same contextual model shown earlier.
Plot the estimated probabilities for category 5 obtained in Exercise 11 by using the marginsplot command.
Use the margins command to calculate the estimated probabilities of being in category 5 (i.e., Y=5) for gender when holding other predictor variables at their means.
Use the margins command to calculate the estimated probabilities of being in category 2 (i.e., Y= 2) for the predictor variable gender when holding other predictor variables at their means.
Use the estimates stats command to compute the AIC and BIC statistics for the three fitted models in Exercises 1, 2, and 5. Interpret the results. Do they support the finding from the log likelihood ratio tests?
Conduct log likelihood ratio tests among the three fitted models. Which one fits the data best? Why?
Interpret the odds ratio of urban in the contextual model.
Write level 1 and level 2 equations for the contextual model shown in Exercise 5.
Fit a two-level, contextual model with gender, byses, and usecalc as the student- level predictor variables and urban as the school-level predictor variable. Specify a random intercept and a random slope for usecalc. No cross-level interactions are included.
Conduct a likelihood ratio test comparing the random-intercept model and the unconditional model. Interpret the results.

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Applied Ordinal Logistic Regression Using Stata 1st Edition Xing Liu - Solutions

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