SPRING Final Exam. May 5, 1999

Transcription

1 IE 230: PROBABILITY AND STATISTICS IN ENGINEERING SPRING 1999 Final Exam May 5, 1999 NAME: Show all your work. Make sure that any notation you use is clear and well deæned. For example, if you use a new random variable X or an event A, you must tell me what X or A represents. If your notation is not clearly deæned you will lose points, even if your answer is correct. You do not need to simplify numeric answers, e.g., 2æ9 is an acceptable ænal 36 answer.

2 Problem 1 è6 pointsè: A machine contains three components. Let T i be the time until failure in hours èi.e., the lifetimeè of component i, for i =1; 2; 3. Assume the machines are independent and that each T i is an exponential random variable with rate ç =0:01. The machine will run as long as all three components are working, and will fail if any of the three components fail. Find the probability that the machine runs for at least 100 hours.

3 Problem 2 è6 pointsè: A shipping company handles boxes of three diæerent types. Boxes of type1have avolume of 8 cubic feet, boxes of type2have avolume of 27 cubic feet, and boxes of type3have avolume of 64 cubic feet. Let X i denote the number of boxes of type i shipped during a given week, for i =1; 2; 3. The X i are independent and: EëX 1 ë = 20 V ëx 1 ë=10 EëX 2 ë = 25 V ëx 2 ë=14 EëX 3 ë = 10 V ëx 3 ë=6: a: Let Y = the total volume shipped in a given week. Write Y as a function of the X i, i =1; 2; 3. b: Find the expected value of Y. c: Find the variance of Y.

4 Problem 3 è8 pointsè: Suppose exam scores have a mean of 70 and a standard deviation of 8. I give an exam to 100 randomly selected students. What is the probability that the average exam score for these 100 students is greater than 72? Show your work.

5 Problem 4 è9 pointsè: The lifetime, in hours, of a lightbulb is an exponential random variable with parameter ç. To estimate ç, I take a random sample of 5 light bulbs and measure the life, in hours, of each bulb. Let x 1 =6;x 2 =4;x 3 =2;x 4 =5;x 5 = 3 be the observed lifetimes. Find the maximum likelihood estimate of ç. Recall: Lèçè =f X èx 1 ;çè æææf X èx n ;çè. Show your work.

6 Problem 5 è9 pointsè: A certain brand of soft drink, called brand C, has been advertising heavily on television. When making a purchase, a customer who has seen the advertising will choose brand C 75è of the time. A customer who has not seen the advertising will choose brand C 50è of the time. Assume that 60è of all customers have seen the advertising. A customer enters a store and chooses brand C. What is the probability that this customer has seen the advertising for brand C?

7 Problem 6 è9 pointsè: Suppose I own a plant that produces electrical components and 5è of all components produced at my plant are defective. Assume components are independent. a: A customer orders a batch of 10 components. Find the probability that at least 9 of these 10 components are not defective. b: I send a batch of 10 components to a customer. What is the expected number of defective components in the batch? c: I need to ænd a defective component. I will repeatedly select components at random until I have found one that is defective, then I will stop. What is the probability I must select more than two times?

8 Problem 7 è12 pointsè: Suppose X and Y are two continuous random variables with density function f XY èx; yè=xy for 0 ç x ç 2 and 0 ç y ç 1. a: Find the marginal probability density function for X, f X èxè. b: Find P èy ç 0:5jX = 1è. c: Are X and Y independent? Prove why or why not. d: What is the correlation between X and Y, ç XY?

9 Problem 8 è9 pointsè: Suppose customers arrive at a bank in a Poisson process with average arrival rate of two customers per minute. a: What is the probability that at most two customers arrive in the next 5 minutes? b: How long do I need to wait, on average, for the ærst 10 customers to arrive? c: What is the probability that I must wait at least 5 minutes for the next customer to arrive?

10 Problem 9 è8 pointsè: A pharmaceutical manufacturer is advertising a new weight loss drug which they claim results in an average weight loss of 9 pounds in one month. To test this claim, a random sample of 4 subjects was chosen to use the product for one month. For each subject, we record the total weight loss in pounds. The data are as follows: Does the data provide any evidence to disprove the manufacturer's claim? Use a 90è conædence level. Assume weight loss is normally distributed.

11 Problem 10 è6 pointsè: Heights èmeasured in inchesè of students at Purdue are normally distributed with a mean ç, unknown, and variance ç 2 = 36. I want to obtain a 92è conædence interval for ç that has a total length of no more than 2 inches. What sample size is required?

12 Problem 11 è18 pointsè: Suppose we are interested in determining whether a higher proportion of males smoke cigarettes than females. Let p 1 be the probability that a randomly selected male smokes. Let p 2 be the probability that a randomly selected female smokes. We want to estimate ç = p 1, p 2. To do this, we take a random sample of n 1 males and n 2 females and ask each person whether or not they smoke. Assume that whether or not an individual smokes is independent of other individuals. a: Deæne, very clearly and in words, some appropriate random variables for this situation. b: Using the random variables deæned in part a, what would be a reasonable estimator of ç? c: Let ^æ be the estimator given in part b. Is^æ unbiased? Prove whyorwhy not. d: What is the standard error of ^æ? Recall that seè ^æè = q V è ^æè.

13 e: How would you estimate the standard error of ^æ? f: What is the mean squared error of ^æ? Recall that MSEè ^æè =Varè ^æè+èbiasè 2.

14 Extra Credit è4 pointsè: This question is for extra credit. It will be graded very stringently, i.e., very little partial credit will be given. a: è2 pointsè In problem 11, how would you form a 95è conædence interval for ç? Hint: What kind of random variable is ^æ? b: è2 pointsè The random variables X 1 and X 2 have a joint probability density function f X1 ;X 2 èx 1 ;x 2 è. Prove that where a 1 and a 2 are constants. Eëa 1 X 1 + a 2 X 2 ë=a 1 EëX 1 ë+a 2 EëX 2 ë;

15 FORMULAS: If X 1 ;:::;X n are random variables and a 1 ;:::;a n are constants, then Eëa 1 X 1 + æææ+ a n X n ë=a 1 EëX 1 ë+æææ+ a n EëX n ë: If X 1 ;:::;X n are independent random variables and a 1 ;:::;a n are constants, then V ëa 1 X 1 + æææ+ a n X n ë=a 2 1V ëx 1 ë+æææ+ a 2 nv ëx n ë: P èa or Bè =P èaè+p èbè, P èa and Bè P èajbè = P èa and Bè P èbè If A 1 ;:::;A K are mutually exclusive and exhaustiveevents, then P èbè = P k i=1 P èbja ièp èa i è Conædence Intervals Problem Type Point Estimate Two Sided 100è1, æèè Conædence Interval CI for ç with ç 2 known x x, z æ=2 ç= p n ç ç ç x + z æ=2 ç= p n CI for ç with ç 2 unknown x x, t æ=2;n,1 s= p n ç ç ç x + t æ=2;n,1 s= p n for normal population Discrete Distributions Distribution f X èxè =P èx = xè EëXë VëXë 1 Uniform èa; bè ; x = a; a +1;:::;b a+b èb,a+1è 2,1, b,a Binomial èn; pè n æ px è1, pè n,x ; x =0; 1;:::;n np npè1, pè x Geometricèpè ç pè1, pè x,1 ; x =1; 2;::: 1=p è1, pè=p 2 Negative Binomèr;pè p r è1, pè x,r ; x = r;r+1;::: r=p rè1, pè=p 2 ç x,1 r,1 è HypergeomèN; K; nè K x èè N,K n,x è ; x =1;:::;minèn; Kè np npè1, pè èn,nè è N n è, p = K=N èn,1è e Poissonèçè,ç ç x ; x =0; 1;::: ç ç x! n! Multinomial x 1!æææx k! px 1 1 æææpx k k EëX i ë= V ëx i ë= P èn; p 1 ;:::;p k è k x i=1 i = n; P k p i=1 i =1 np i np i è1, p i è Continuous Distributions Distribution f X èxè F X èxè =P èx ç xè EëXë VëXë 1 Uniformèa; bè ; a ç x ç b x,a ; a ç x ç b a+b b,a b,a 2 12 Exponentialèçè çe,çx ; x ç 0 1, e,çx ; x ç 0 1=ç 1=ç 2 Erlangèr;çè Normalèç; ç 2 1 è p ç r x r,1 e,çx e,èx,çè 2 2çç èr,1è! ; x ç 0 R x 0 èb,aè 2 ç r u r,1 e,çu èr,1è! du; x ç 0 r=ç r=ç 2 2ç 2 ;,1 éxé1 See Tables Below ç ç 2

16 Property Discrete Random Variables Continuous Random Variables Marg Distribs f X èxè =P èx = xè f X èxè Event Probs P èx 2 Aè = P x2a f X èxè P èx 2 Aè = R x2a f Xèxèdx Exp Val EëXë= P x xf X èxè EëXë= R x xf Xèxèdx Exp Val of Fn EëgèXèë = P x gèxèf Xèxè EëgèXèë = R x gèxèf Xèxèdx Var V ëxë= P xèx, ç X è 2 f X èxè V ëxë= R x èx, ç Xè 2 f X èxèdx V ëxë=eëx 2 ë, EëXë 2 V ëxë=eëx 2 ë, EëXë 2 Joint Distribs f XY èx; yè=p èx = x and Y = yè f XY èx; yè f XY Z èx; y; zè=p èx = x; Y = y; Z = zè f XY èx; yè= P z f XY Z èx; y; zè f XY Z èx; y; zè f XY èx; yè= R z f XY Zèx; y; zèdz Event Probs P èèx; Y è 2 Aè = P èx;yè2a f XY èx; yè P èèx; Y è 2 Aè = R èx;yè2a f XY èx; yèdxdy Marg Distribs f X èxè = P y f XY èx; yè f X èxè = R y f XY èx; yèdy Cond Distribs f Y èyè = P x f XY èx; yè f X èxè = P P y z f XY Z èx; y; zè f Y jx=x èyè = f XY èx;yè f X èxè f XY jz=z èx; yè= f XY Zèx;y;zè f Z èzè f XjY =y;z=z èxè = f XY Zèx;y;zè f YZ èy;zè f Y èyè = R x f XY èx; yèdx f X èxè = R R y z f XY Zèx; y; zèdydz f Y jx=x èyè = f XY èx;yè f X èxè f XY jz=z èx; yè= f XY Zèx;y;zè f Z èzè f XjY =y;z=z èxè = f XY Zèx;y;zè f YZ èy;zè X,Y Indep f Y jx=x èyè =f Y èyè f Y jx=x èyè =f Y èyè f XY èx; yè=f X èxèf Y èyè f XY èx; yè=f X èxèf Y èyè X,Y,Z Indep f XY Z èx; y; zè=f X èxèf Y èyèf Z èzè f XY Z èx; y; zè=f X èxèf Y èyèf Z èzè Covariance ç XY = EëèX, EëXëèèY, EëY ëèë ç XY = EëèX, EëXëèèY, EëY ëèë Correlation ç XY = EëXYë, EëXëEëY ë ç XY = EëXYë, EëXëEëY ë EëXYë= P P x y xyf XY èx; yè EëXYë= R R x y xyf XY èx; yèdy dx ç XY = ç XY ç X ç Y ç XY = ç XY ç X ç Y