Math 29 Probability Final Exam. Saturday December 18, 2-5 pm in Merrill 03

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1 Name: Math 29 Probability Final Exam Saturday December 18, 2-5 pm in Merrill 03 Instructions: 1. Show all work. You may receive partial credit for partially completed problems. 2. You may use calculators and a two-sided sheet of reference notes. You may not use any other references or any texts, except the provided z-table. 3. You may not discuss the exam with anyone but me. 4. Suggestion: Read all questions before beginning and complete the ones you know best first. Point values per problem are displayed below if that helps you allocate your time among problems. 5. You need to demonstrate that you can solve all integrals in problems that do not have a (DO NOT SOLVE) statement. I.E. write out some work showing how you solved the integration, including if necessary integration by parts. 6. Probabilities should be given as NUMERICAL values, unless I say an expression is warranted. 7. If the next part of a problem depends on a previous part which you cannot solve, PICK a distribution to use, so you can get partial credit. 8. Good luck! Problem Total Points Earned Possible Points

2 1. Let X denote the number of solo travelers who arrive at a particular free coffee stop (sponsored by local Boy and Girl scouts) at a rest stop along I-91 in an hour. Suppose it is known an average of 16 solo travelers arrive each hour, and that these travelers are independent of one another (note we are looking at solo travelers, not families, so this makes sense), and hours are also independent time intervals. a. What is the probability the coffee stop at the rest stop sees only 8 or 9 solo travelers in a half hour? b. What distribution would you use to model how long the Boy and Girl scouts have to wait for the next solo traveler after one has just arrived? (Be specific with parameter values, and time unit). c. What is the probability the Boy and Girl scouts wait for longer than 5 minutes for the next solo traveler after one has just arrived?

3 2. Suppose X 1 is Gamma(2,3) and X 2 is Gamma(3,2), and those 2 random variables are independent. Let Y 2X 1 3X 2. a. Using an appropriate method, determine the distribution of Y and show your work as justification. b. Suppose you have 50 independent random variables that behave like Y (i.e. have the same distribution), and you construct their average, call it Y 50. What numerical value does Y50 converge in probability to? Justify your answer.

4 3. Wendy s proclaims that there are 256 different ways to order a hamburger. For each hamburger, you have your choice of 8 different condiments (cheese, mayo, pickles, etc.), which can be added to the hamburger or left off (default). Assume that you cannot order extra condiments (i.e. your only options are each condiment on or off the hamburger). a. Show that Wendy s claim of 256 different ways to order a hamburger is correct. b. Suppose you really want all 8 condiments on your burger, but the local Wendy s is low on supplies, so they say you can only pick 3 condiments for your burger. How many different burgers can you order? c. Suppose that Wendy s #136 and Wendy s #202 compete for local business with Wendy s #136 getting 30% of local Wendy s business (you can assume this percentage holds for just burger business). Suppose that 30% of all burgers sold at Wendy s #136 have bacon on them, while at Wendy s #202 that percentage is only 15%. If a friend offered you a Wendy s burger with bacon on it, what is the probability your friend went to Wendy s #202 to get that burger?

5 4. A Little Theory, Variance, and Matching a. Suppose X and Y are jointly distributed random variables with joint pdf f(x,y), with marginal pdfs and conditional pdfs in our usual notation. Show that E( XY) E[ XE( Y X )]. b. Suppose X and Y are jointly distributed random variables with correlation -0.4, Var(X)=4 and Var(Y)=36. What is Var(4Y-3X)? c. Matching. (Not all choices may be used). Having a random sample implies this about the associated random variables A stochastic process where the variables are related by conditional expectations The minimum of a random sample is an example Result related to convergence in distribution A. Convergence in quadratic mean B. Poisson process C. Markov Chain D. Martingale E. I.I.D. F. CLT (Central Limit Theorem) G. Dependence H. Order statistics I. Standardization Requires stationarity and independence

6 5. Suppose (X,Y) is a point chosen on the unit square with probability governed by the joint pdf: f ( x, y) x y,0 x 1,0 y 1, and 0, otherwise. a. What is the pdf of Z, which denotes the area of the rectangle formed by (0,0), (x,0), (0,y), and (x,y)? b. In a one-dimensional setting, the method of is just a faster way of doing the method of when the inverse function is 1-1, in the context of transformations.

7 6. An Amherst College student owns 2 umbrellas and takes them back and forth between her room and library carrel. Let X t denote the number of umbrellas in her room at the start of day t. Suppose the student only carries an umbrella with her if it is currently raining, and only if one is available at her present location. She also only makes a trip to her carrel and back to her room once a day. Finally, assume the probability it is raining when she leaves her room or library carrel is.4, and is independent of all past weather activity a. Complete the Markov Chain transition matrix to describe the behavior of X t.recall rows = starting state, columns = ending state b. Does your Markov chain have any absorbing sets apart from the entire sample space? Yes No Is your transition matrix reducible or irreducible? Reducible Irreducible c. What is the probability there will be no umbrellas available in the student s room at the start of the day 2 days from now if there is only one available today? (i.e. 1 today, tomorrow= whatever, third day=0 umbrellas) d. Set up a system of equations (DO NOT SOLVE) that would allow you to find the long run probability the student gets wet (no umbrella and it is raining) in the morning when she leaves her room.

8 7. Suppose X and Y are jointly distributed random variables with joint pdf f ( x, y) 8xy,0 x y 1, and 0, otherwise. a. Sketch and shade the region where the joint pdf has positive density on the graph paper at right. b. Set up an integral or integrals(do NOT SOLVE) to find P(3X<Y,Y<.75). c. Compute Cov(X,Y). d. Based on your computation in part c., can you conclude that X and Y are dependent? Yes No

9 8. Lifetimes of certain light bulbs are well-modeled by a distribution with mean 102 hours and variance 64 hours, with individual bulbs having independent lifetimes. A company needs 36 bulbs for their new office. They will use the bulbs and sign a lifetime contract to continue that use if the mean lifetime of the first 36 bulbs is greater than 100 hours. What is the probability the company signs a lifetime contract to use these bulbs? 9. A new version of a drug test can detect the drug in subjects using the drug 85% of the time. However, it also detects the drug in 10% of people who do not use the drug. If the percentage of drug users in the population is.1%, provide a probabilistic argument against a company policy that will immediately fire anyone who is detected as using the drug (i.e. no additional tests).

10 10. Let X and Y be jointly distributed random variables with joint pdf given by f ( x, y) 2( x y),0 x y 1, and 0, otherwise. a. Find the marginal pdf of Y. b. Find the general conditional pdf of X given Y. c. Find E(X Y=.5). d. Set up an integral (DO NOT SOLVE) to find P(X>.25 Y=.75)