Assignmnets Ie255 Fall 2005

ASSIGNMNETS IE255 – FALL 2005 (All problems must be prepared at home. Those with (*) will be handed in and will be controlled.)

PS 1: Probability (due: 5.10. at 9.00)

1) A college plays 12 football games during a season. In how many ways can the team end the season with 6 wins, 4 losses, and 2 ties?

2) 11 people are going on a skiing trip in 3 cars that hold 4, 4, and 5 passengers, respectively. In how many ways is it possible to transport the 9 people to the ski lodge, using all cars?

3) (*) If 3 books are picked at random from a shelf containing 5 novels, 4 books of poems, and a dictionary: a) What is the probability that the dictionary is selected? b) 2 novels and 1 book of poems are selected?

4) (*) For married couples living in a certain suburb the probability that the husband will vote on a bond referendum is 0.25, the probability that his wife will vote in the referendum is 0.32, and the probability that both the husband and wife will vote is 0.15. What is the probability that a) At least one member of a married couple will vote? b) A wife will vote, given that her husband will vote? c) A husband will vote, given that his wife does not vote? d) Are the events “the wife will vote” and “the husband will vote” independent?

5) (*)Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 25% of the packages, fails to stamp the expiration date once in every 250 packages; Tom, who stamps 40% of the packages, fails to stamp the expiration date once in every 100 packages; Jeff, who stamps 25% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 10% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of film does not show the expiration date, what is the probability that it was inspected by John?

6) (*)A truth serum has the property that 90% of the guilty suspects are properly judged while, of course, 10% of guilty suspects are improperly found innocent. On the other hand, innocent suspects are misjudged 2% of the time. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the prob. that he is innocent?

7) From a group of 5 men and 5 women, how many committees of size 3 are possible. a) With no restrictions? b) With 1 man and 2 women? c) With 2 men and 1 woman if a certain man must be on the committee?

8) (*)The probability that a patient recovers from a delicate heart operation is 0.85. What is the probability that a) Exactly 2 of the next 3 patients who have this operation survive? b) All of the next 3 patients who have this operation survive? c) What is a necessary assumption that we can solve part a) and b) of this question.

9) In a certain federal prison it is known that 2/3 of the inmates are under 25 years of age. It is also known that 3/5 of the inmates are male and that 5/8 of the inmates are female or 25 years of age or older. What is the probability that a prisoner selected at random from this prison is female and at least 25 years old?

10) (*)From a box containing 4 black balls, 3 red balls and 3 green balls. 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. What is the probability that a) All 3 are the same colour? b) Each color is represented?

11) (*)From a box containing 4 black balls, 3 red balls and 3 green balls. 3 balls are drawn in succession without replacement. What is the probabilty that the third ball drawn is black?

IE-255: PS 2: Random variables(due: 12.10. at 9.00)

12) (*) In a card game out of 52 cards (4 of them aces) a single card is handed to each of 4 players. You are the fourth player. a) What is the probability that you get an ace? b) What is the probability that you get an ace if you have seen by chance that the first player got an ace? c) You get an ace. What is the probability that all others have not got an ace?

13) (*)From a box containing 4 black balls, 3 red balls and 3 green balls. 3 balls are drawn with replacement.

a) Find the probability function f(x) of the random variate X = “Sum of red balls drawn”. b) Calculate F(2) and F(0.17).

14) From a box containing 4 black balls, 3 red balls and 3 green balls. 3 balls are drawn without replacement.

a) Find the probability function f(x) of the random variate X = “Sum of red balls drawn”. b) Calculate F(2) and F(0.17).

15) (*) The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution 0 x ≤ 0 F(x)= 1 − e − 3 x x > 0

Find the probability of waiting less than 12 minutes between successive speeders c) using the cumulative distribution of X; d) using the probability density function of X.

16) (*)Consider the density function kx 0 < x < 4 f(x)= 0 elsewhere a) Evaluate k, b) Find F(x) and use it to evaluate P(0.3 < X < 0.6).

17*) A continuous random variable has density 1+x for -1 < x ≤ 0 f(x) = 0.5 for 0 < x ≤ 1 0 else a) Check that f is a density. b) Find the CDF F(x). c) Compute the probability P(- 0.2 < X ≤ 0.5).

18) Consider the (continuous) random variate X=”time in minutes that it takes a randomly selected student of our class to solve question 10”. Assume that the time is always between 0.5 and 10 minutes and most students work about 3 minutes. Construct a function that can be used as density for the random variate X. IE-255: PS 3: Multivariate Random variables, Mean, Variance, Conditional Expectation (due: 19.10. at 9.00)

19) (*) 3 balls are selected at random from an urn with 3 blue, 2 red and 1 green ball. Let X be the total number of blue balls selected and let Y be the number of red balls selected. a) Find the joint probability function f(x,y). b) Find the marginal distribution of X. c) Find the conditional distribution of Y given that X is equal to 1. d) Are X and Y independent?

20) (*) For two random variates X and Y the joint density function is:

24xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x+y ≤ 1 f(x,y)= 0, elsewhere. a) Find the probability that X +Y > ½ . b) Find the marginal density for Y . c) Find the conditional density of Y given that X = 0.5 . d) Find the probability that Y is less than 1/8 if it is known that X is 0.5 . e) Are X and Y independent?

21) Consider the two random variables X and Y and their joint density: 4xy, 0 < x < 1; 0 < y < 1 f(x,y) = 0, elsewhere. a) Check if f is a density. b) P(0 ≤ X ≤ 1/2 and ¼ ≤ Y ≤ 1/2); c) P(X

22) (*) Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 psi. Let X denote the actual air pressure for the right tire and Y denote the actual air pressure for the left tire. Suppose that X and Y are random variables with the joint density k(x2+y2), 30 ≤ x ≤ 50; 30 ≤ y ≤ 50 f(x,y) = 0, elsewhere

a)Find k. b)Find P(30 ≤ X ≤ 40 and 40 ≤ Y ≤ 50). c)Find the probability that both tires are underfilled. d) Are X and Y independent?

23.* The density function of the continous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year, is given as x, 0

24) A continuous random variable has density 1+x for -1 < x ≤ 0 f(x) = 1 for 0 < x ≤ 0.5 0 else a) Check that f is a density. b) Find the CDF F(x). c) Compute the expectation and variance of X. d) Find the conditional density of X: f(x|X < -0.5) e) Find the conditional expectation of X: E(X| X < -0.5)

25) * Suppose that X and Y have the following joint density: f(x,y) = 2 for 0

2 2 30) If X and Y are independent random variables with expectations μx = 1 and μy = 2 and variances  x=5 and  y=3, find the expectation and variance of the random variable Z= -2X + 4Y - 3.

31*) Repeat question 30 for the case that X and Y are not independent and xy = 1. 32) Compute P(-2< X <+2), where X has the density function 6x(1-x), 0

PS 6:Uniform, Normal, Exponential and Gamma distribution, quantiles (due: 9.11. at 9.00 )

40*) In the November 1990 issue of Chemical Engineering Progress a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.63 with a standard deviation of 0.08. Assume that the distribution of percent purity was approximately normal. a) What percentage of the purity values would you expect to be between 99.6 and 99.7? b) What percentage of the purity values is more than 0.1 away from the mean? c) What purity value would you expect to exceed exactly 7% of the population? d) What purity value is exceeded by exactly 3% of the population?

41*) The weights of a large number of miniature poodles are approximately normally distributed with a mean of 9 kilograms and a standard deviation of 0.7 kilogram. If measurements are recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights a) over 9.5 kilograms; b) at most 8.6 kilograms; c) between 7.3 and 9.1 kilograms inclusive; d) of 9 kg.

42*) A bus arrives every 13 minutes at a bus stop. It is assumed that the waiting time for a particular individual is a random variable with a uniform distribution. a) What is the probability that the individual waits more than 7 minutes? b) What is the probability that the individual waits between 2 and 7 minutes?

43*) Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 20 drivers on the road is drunk. If 1500 drivers are randomly checked next Saturday night, what is the probability that the number of drunk drivers will be at least 70 but less than 94?

44*) In a biomedical research activity it was determined that the survival time in weeks of an animal when subjected to a certain exposure of gamma radiation has a gamma distribution with = 5 and =10. a) What is the mean survival time of a randomly selected animal of the type used in the experiment? b) What is the standard deviation of survival time? c) What is the probability that an animal survives more than 30 weeks?

45*) We assume that the lifetime of an electric bulb follows an exponential distribution with mean value 5000 hours. Find the lifetimes that are exceeded by the probabilities 50% and 5%.

Computer Assignment 1 – 3) (due 15.11) 1*) In one hour 5000 cars are passing a certain filling station on a high-way. We assume that a single driver decides (independently of the others) to enter the station with a probability of 0.02. a) Compute the exact probability that less than 77 cars enter the station. b) Compute the probability that between 90 and 110 cars are entering the station. c) Compare the exact result of a) and b) with the approximate result using the Poisson distributions as approximation. d) As c) but with the normal distribution instead of the Poisson distrib. (With and without continuity correction.)

2*) Among 2000 electric devices 200 are defective. For quality control reasons 100 randomly chosen pieces are tested. a) Compute the exact probability that less than 4 of the selected 100 pieces are defective. b) Compute the exact probability that between 15 and 20 pieces are defective. c) Compare the exact result of a) and b) with the approximate result using the Binomial distribution as approximation for the correct distribution. d) As c) but with the normal distribution instead of the Binomial. (With and without continuity correction.)

3*) A bookshop is selling a certain monthly journal for 5$ per piece and buys it from a publishing house at 3$ per piece. Lets assume that the number of sold journals per month is Poisson distributed with expectation 120. As the bookshop cannot hand back journals that were not sold, it has to decide about the number of journals that are ordered every month. a) Compute the expectation and the variance of the money the bookshop gains when ordering 100 journals. b) Compute the expectation and the variance of the money the bookshop gains when ordering 120 journals. c) Try to find the number of ordered journals that maximises the expected gained money. Compute the expectation and variance of the gained money for that number of orders. d) Comment on the interpretation of the variance in this example. What will the manager of the bookshop try to do if he does not want to take too much risk.

Solve all computer assignments with EXCEL in one file with several sheets. Create a zip file of that file and use your name as filename. E-mail your file to guven.demirel @boun.edu.tr till 15.11.2005 at 24.00. PS 7: Gamma distributions, Poisson-process, life-time distributions and hazard rate(due: 16.11. at 9.00)

46*) The lifetime in weeks of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation 50 weeks. a) What is the probability that the transistor will last at most 50 weeks? b) What is the probability that the transistor will not survive the first 10 weeks?

47*) The life of a certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant and the exponential distribution applies. a) What is the mean time to failure? b) What is the probability that 200 hours will pass before a failure is observed?

*48) We are given a Poisson process with rate λ = 0.1 per hour. a) What is the distribution and the density of the waiting time till the first event occurs. b) What is the distribution and the density of the waiting time till the 3rd event occurs.

49) Proof that the sum of three independent exponential random variates (all have mean 1) is gamma-distributed with parameters α = 3 and β = 1. 50) a) Compute the mean and the variance of the exponential distribution with (λ = 1). b) Use the result of a) and the result of exercise 49) to calculate the mean and the variance of the Gamma- distribution with (α = 3; β = 1). *51) Assume that the lifetime of a technical part is uniformly distributed between 0 and 5 years. a) Find the failure rate (=hazard rate) function. b) Given that the part worked for 3 years. What is the probability that it fails in the next week? Compute this probability exact and approximately using the result of a). c) Compute the expected left life-time: E( X | X > 3) – 3 c) Given that the part worked for 1 year. What is the probability that it fails in the next week? Compute this probability exact and approximately using the result of a). d) Compute the expected left life-time: E( X | X > 1) – 1 PS 8: Transformation theorem and Moment Generating function (due: 23.11. 9.00)

*52) Let X be a binomial random variable with n=3 and p=1/2. Find the probability distribution of the random variable Y=X2 *53) Let X have a continuous uniform distribution between 0 and 1. Show that the random variable Y= – 2 lnX has a Gamma distribution. Find the parameters of the Gamma distribution. 54) A dealer’s profit, in units of $1000, on a new automobile is given by Y=X1/2, where X is a random variable having the density function 2(1-x), 0

*55) Let X be a random variable with probability distribution f(x) = (1+x)/2, -1

56) The random variable X has density f(x) = 2 – 2x for 0 < x < 1 0 else a) Compute the density of Y = α + β X for arbitrary α and β > 0. b) Compute mean and variance for Y. c) Make plots of the density for different choices of α and β. Explain, why α is called location and β is called scale parameter.

*57) A random variable X has the discrete uniform distribution f(x)= 1/k for x=1,2,3,...,k and 0, elsewhere. t kt t Show that the moment-generating function of X is Mx(t)= e (1-e )/k(1-e ).

*58) A random variable X has the geometric distribution g(x;p)= pqx-1 for x=1,2,3,... t t a) Show that the moment-generating function of X is Mx(t)= pe /(1-qe ) b) Use Mx(t) to find the mean and variance of the geometric distribution.

59) A random variable X has the Poisson distribution p(x;μ)=e-μμx/x! for x=1,2,3,... μ( t a) Show that the moment-generating function of X is Mx(t)= e e -1). b) Using Mx(t), find the mean and variance of the Poisson distribution.

*60) Use the result of 59) to prove that the sum of two independent Poisson random variables is again Poisson distributed. PS 9: Distribution of X+Y , multi-normal distribution Sampling theorems: (due: 30.11. )

61) X ~ N( 20; σ = 3 ) and Y ~ N( 15; σ = 2 ), X and Y independent: Compute the probability P(Y>X).

*62) X ~ N( 12; σ = 3 ); Compute the probability that the sum of 10 independent realisations of X is bigger than 150.

63) X ~ N( 20; σ = 3 ) and Y ~ N( 15; σ = 2 ), X and Y independent: a) Compute the probability that P(2X+3Y> 80). b) Compare the probability of 2 X > 50 and X + X > 50.

*64) Assume that the random variables X and Y describe the distribution of the price of two different stocks a month in the future. We assume that X ~ N( 17; σ = 3 ) and Y ~ N( 12; σ = 2 ). a) If X and Y are independent compute the probability that X+Y is smaller than 25. b) If X and Y are joint normal and have ρXY=0.5 compute the probability that X+Y < 25. Hint: Remember that Cov(X Y) = ρXY σY σX c) For X, Y joint normal and ρXY=0.5 we consider the random variate S=2X+3Y . (S is the value of a portfolio with two stocks of Company X and 3 of company Y. Compute the value that S exceeds with probability 95%. Remark: c) could be seen as a “worst case analysis” for the value of the portfolio and is linked to the “value at risk” concept.

65) If the standard deviation of the mean for the sampling distribution of random samples of size 36 from a large or infinite population is 2, how large must the size of the sample become if the standard deviation is to be reduced to 1.2?

66*) A soft-drink machine is being regulated so that the amount drink dispensed averages 240 milliliters with a standard deviation of 16 milliliters. Periodically, the machine is checked by taking a ssample of 144 drinks and computing the average content. The company official found the mean of 144 drinks to be 236 milliliters and concluded that the machine needed no adjustment. Was this a reasonable decision?

67) The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean μ=3.2 minutes and a standard deviation σ = 3 minutes. If a random sample of 81 customers is observed, find the probability that their mean time at the teller’s counter is more than 3.5 minutes.

68*) The mean score for freshmen on an aptitude test, at a certain college, is 540, with a standard deviation of 50. What is the probability that two groups of students selected at random, consisting of 32 and 50 students, respectively, will differ in their mean scores by an amount between 5 and 10 points? Assume the means to be measured to any degree of accuracy.

69*) Find the probability that a random sample of 25 observations, from a normal population with variance σ2=6, will have a variance s2 a) greater than 9.1 b) between 3.462 and 10.745. Assume the sample variances to be continuous measurements.

70*) A normal population with unknown variance has a mean of 21. Is one likely to obtain a random sample of size 16 from this population with a mean of 24 and a standard deviation of 4.1? If not, what conclusion would you draw?

71) If X1, X2,..., Xn are independent random variables having identical exponential distributions with parameter θ, show that the density function of therandom variable Y= X1+ X2+...+ Xn is that of a gamma distribution with parameters α = n and β = θ. PS 10: Confidence and tolerance intervals for the mean + CI for difference of means (due: 14.12.)

2 2 72*) If S1 and S2 represent the variances of independent random samples of size n1=8 and n2= 12, taken from normal 2 2 populations with equal variances, find the P(S1 /S2 < 4.89).

2 2 73) If S1 and S2 represent the variances of independent random samples of size n1=25 and n2= 31, taken from normal 2 2 2 2 populations with variances σ1 =10 and σ2 =15, respectively, find the P(S1 /S2 > 1.26).

74*) A random sample of 100 car owners shows, that in Virginia, a car is driven on the average 20,500 km per year, with a standard deviation of 4000 km. (a) Construct a 99% confidence interval. (b) Explain the result of a) in a sentence. (c) What can we say about the possible size of error, if we estimate the mean kilometers per year as 20,500?

75*) a) Referring to exercise 74) construct a 99% tolerance interval of the kilometers travelled by cars annual in Virginia. b) Explain the result of a) in a sentence.

76*) A Taxi company is trying to decide whether to buy brand A or brand B tires for its cars. A random experiment is conducted using 12 of each brand. The number of kilometers is recorded till the tires wear out. For brand A we obtain: Sample mean 36,300. sample standard deviation 3500. For brand B we obtain: Sample mean 38,100. sample standard deviation 5000. a) Compute a 99% confidence for the difference of the two means. (Do not assume equal variance.) b) Which assumptions are necessary?

77*) In a different experiment for 8 taxis a brand A and a brand B tire a randomly assigned to the rear wheels. The results are Taxi Brand A Brand B 1 36,000 36,200 2 45,500 46,800 3 36,700 37,700 4 32,000 31,100 5 48,400 47,800 6 32,800 36,400 7 38,100 38,900 8 30,100 31,500 a) Find a 95% confidence interval for the difference of the two means. b) Which assumptions are necessary? c) What can the company learn from the result. What should the manager do? d) In General: Which form of the experiment (that of number 76 or 77) do you think is better? Why?

PS 11: Confidence intervals for proportion and variance(due: 21.12)

78*) In a random sample of 1000 homes in a certain city, it is found that 228 are heated by oil. a) Find the 95% confidence interval for the proportion of homes in this city heated by oil. b) What sample size is necessary to obtain a CI that is not longer than 0.01 if we assume that the true porportion is about 0.23. c) What sample size is necessary to obtain a CI that is not longer than 0.01 if we make no assumptions about the proportion.

79*) A manufacturer wants to compare the percentage of Turkish citizens who know his product with this percentage 5 years ago. 5 years ago in a random sample of size 2000 1050 people knew his product. Today of 1000 asked people 570 know his product. a) Compute a 90% CI for the difference of the two proportions. b) How can the manufacturer interpret the CI?

80*) A random sample of size 200 obtained a mean of 72 and a standard deviation of 16. a) Construct a 95% CI for σ. b) What assumptions are necessary?

PS 12: Unbiased estimates, maximum likelihood estimation, moment estimates (due 28.12.)

81) Show that the sample mean is an unbiased estimator for the mean μ of the parent population.

82*) Suppose that there are n trials from a Bernoulli process with parameter p, the probability of success. Work out the maximum likelihood estimators for the parameter p. 83) Consider the normal distribution. a) Develop the maximum likelihood estimator for μ and σ2. b) Develop the moment estimators. 84*) Consider the log-normal distribution. a) Develop the maximum likelihood estimator for μ and σ2. b) Develop the moment estimators.

85*) Consider observations from the gamma distribution. a) Write out the likelihood function and the set of equations, which when numerically solved, give the maximum likelihood estimators for α and β. b) Develop the moment estimators. c) What is the advantage of the moment estimator? What is the advantage of the maximum likelihood estimator?

Computer Assignment 4) (due 25.12 at 24.00) a) Find examples of simple densities that have increasing failure rate. b) Find examples of simple densities that have decreasing failure rate. c) For many technical parts it is realistic to assume a U-shaped failure rate. Why? Find examples of simple densities that have U-shaped failure rate. Plot the failure rates and densities of this example and interpret the results. Please write the formulas of the densities and the failure rates and your interpretation into the excel-sheets. d) Assume that X follows a Gamma distribution with (α = 3.7; β = 12). Find the probability P(2.7 < X < 12.8).

Computer Assignment 5) (due 25.12) File “comp1.txt” contains a sample from a normal population. a) Use these observations to compute the 95% Confidence Intervals for the mean of the population b) Compute a 99 % CI for the variance.

Computer Assignment 6) (due 25.12) The files “comp2a.txt” and “comp2b.txt” contain two independent samples of the same size of two different populations A and B. a) Compute a 99% CI for the difference between the means of the two populations. b) Compute the differences between the two observations and compute a 99% CI for the mean of the differences. c) Compare the result of a) and b).

Computer Assignment 7) (due 25.12) We want to find an answer to the question if the mathematics or the English part of a University placement exam has the better results. Use both methods to compute a CI. a) “paired observations”: Choose randomly 50 students (you find the data in “comp3a.txt”, the first number in each line is the math result, the second the English result of the same student) and compute the differences between their results. The compute the CI for the mean of the differences. b) “independent samples”: Choose randomly the results of 50 students for the mathematics part (you find the data in “comp3b.txt”). Then choose (independently) the results of 50 students of the English part (you find the data in “comp3c.txt”). Compute the CI for the difference of the means. c) Compare the results of a) and b). Which method is better? To compute a 95% CI for the differences between the mean results which method did you expect to give better results? Why?

Explanations for the Computer assignments: You have to get the file makedata.exe. Down load this file from the address: http://www.ie.boun.edu.tr/~hormannw/

Copy the program into an empty directory. Then change into that directory and start the program in a DOS-window by typing “makedata”. When you are asked to, type in your student number and the program will generate your “personal data” with the file names as explained below.

Now use EXCEL to solve the computer assignments. Open the data files and import them into EXCEL. For the files “comp1.txt”, “comp2a.txt” and “comp2b.txt” you will probably have problems with the decimal point (as it is . and not , ). It is easiest to import the numbers as text and then change . to , by the replace command. The data of assignment 7 have no decimal point, but be sure not to overlook that you must get two columns for the file “comp3a.txt” . Then use the below Excel-Functions and the formulas we have learned to compute the CIs. Use no other statistical functions of EXCEL!! Learn the properties of the below functions from the Excel-Help-Function. Count() to compute the number n, Average() to compute the sample mean, Stdev() to compute the sample standard deviation s, Tinv() to compute the t-values you need for the CI Chiinv() to compute the values for the chi-square distribution.

Solve all three assignments in one file with several sheets. Please e-mail your zipped files to guven.demirel @boun.edu.tr Due date is the 25.12. at 24.00.