Stat/Math 360 Probability Final Exam December 17, Show all work. You may receive partial credit for partially completed problems.

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1 Name: Stat/Math 360 Probability Final Exam December 17, 2014 Instructions: 1. Show all work. You may receive partial credit for partially completed problems. 2. You may use calculators and up to three single-sided sheets of reference notes. The distribution information portion of the textbook (Appendix C) and the MGF reference sheet are provided. You may not use any other references or any texts. You may NOT use your cell phone as your calculator. All cell phones and mobile devices must be turned off and put away. 3. You may not discuss the exam with anyone but Professor Wagaman. Uphold the honor code. 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. Problems are spaced out. White space should NOT be taken as an indicator to fill the space. If you need more room to show work, please use the provided scratch paper, and be sure it has your name on it! 6. Initial in the box (right and below) to acknowledge that you have read and understand the instructions. Problem Total Points Earned Possible Points

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3 1. Suppose Y, the number of defects in a yard of fabric, is known to be conditionally distributed (on ) according to a Poisson distribution with parameter 2. Further suppose that is a random variable 1 / 3 whose pdf is given by f ( ) e, 0 and 0, otherwise. 3 a. Identify the distribution of, specifying all parameters. b. Find E(Y). c. Find V(Y).

4 2. Wheelin', Dealin', and Re-Arrangin. Set up computations to find the probabilities below. Do NOT evaluate. a. Suppose that 6 men and 6 women are randomly put into two groups of six to discuss an issue. What is the probability that both groups have the same number of men? b. Suppose you are dealt five cards from a standard 52 card deck (which contains 13 cards from each of 4 suits). What is the probability your hand of five cards contains at least one card from each suit?

5 3. Suppose X is a random variable with pdf given by 2x, 0 <x<1, and 0, otherwise. 4 a. Find E ( X ). b. Find the CDF of X. Be sure it is fully specified. c. Some R code and output is provided. Explain what is being computed here, and how it relates to the computations above. You should explain what the first line is doing in detail, focusing on WHY that is being done. simlist=sqrt(runif(10000)) mean(simlist^4)

6 4. Let X1, X 2, X 3 be a random sample from a Bernoulli distribution with p=2/3. a. We construct a new RV: W X1 X 2 X 3. Provide the MGF for W, and fully identify W's distribution. b. We construct a second new RV: Y X1X 2 X 3. Provide the MGF for Y. 2 c. Use the MGF for Y to find E ( Y ). Your solution must demonstrate how the MGF may be used to compute this expectation.

7 5. Theory and Matching Check. a. Let A, B, and C be events with positive probability defined on the same sample space, S. c c Prove that: P( A B) P( A BC) P( C B) P( A BC ) P( C B). Be sure to present a well-constructed proof with justified assertions.

8 5. continued. b. Matching. For each description below, match the most appropriate selection (using CLEAR capital letters). Not all options may be used. An inequality only for non-negative RVs An algorithm related to MCMC A distribution useful for modelling fractions A distribution related to coupon collecting (collecting a set) Stronger than almost sure convergence A law related to random digits A useful formula for adding independent RVs Continuous distribution with the memoryless property A. Exponential B. Weak LLN C. Benford s D. Law of Total Probability E. Metropolis-Hastings F. Geometric G. Chebyshev s H. Poisson I. Markov s J. Convergence in Distribution K. Pareto L. Convergence in Probability M. Convolution N. Law of Total Expectation O. Jacobian P. Beta Q. Gamma 6. Esther, a statistician, has noted that her mood each day can be categorized as being cheerful, so-so, or gloomy. From past experience, she finds that her mood on any given day only depends on the mood of the previous day. When she is in a cheerful mood on one day, she has a 70% chance of being cheerful the next day, and the chance of being so-so is double that of being gloomy on the next day. When she is in a so-so mood on one day, she has a 40% chance of being cheerful the next day, with equal chances of repeating so-so or being gloomy on the next day. Finally, when she is in a gloomy mood on one day, she has a 20% chance of being cheerful the next day, with equal chances of repeating gloomy or being so-so the next day. a. Fill in the provided partial transition matrix to describe Esther s moods. Be sure you provide a valid transition matrix. Cheerful 0.7 So-So 0.4 Cheerful So-So Gloomy Gloomy 0.2 b. Esther s mood is cheerful today. Taking that into account, what is the probability she will be cheerful two days from now?

9 6. continued. c. Based on Esther s observations of her mood, what proportion of time is she in a cheerful mood? Find the equilibrium stationary distribution for this Markov Chain to answer this.

10 7. Suppose that X and Y are jointly distributed continuous random variables with joint pdf given by 8 f ( x, y) xy,0 x 1, x y 2x, and 0, otherwise. 3 a. Find the marginal pdf of X. b. Set up computations to find P(Y<1). Do NOT evaluate. c. Find the conditional pdf of Y given X.

11 7. continued. d. Find P(Y>1 X=0.75). e. Find Cov(X,Y).

12 8. An agency employs two administrative assistants who are responsible for sending out all correspondence from the agency. For letter correspondence, the first assistant makes an average of 3 errors per letter when typing, while the second assistant makes an average of 4.2 errors per letter when typing. The assistants have an equal chance of typing each letter. Answer the questions below, using reasonable models for X 1, the number of errors in a letter typed by the first assistant, and for X 2, the number of errors in a letter typed by the second assistant. a. What is the probability a letter typed by the first assistant has exactly two errors? b. What is the probability a letter typed by the second assistant has more than one error? c. Recall that your letter is equally likely to be typed by either assistant. You receive a letter from the agency. What is the probability that it will have no errors?

13 9. In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from -2.5 years to 2.5 years. The healthcare data are based on a random sample of 48 people. Provide R code to compute the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages. You must show your work in order to obtain credit.

14 y Suppose Y is a random variable with pdf given by f ( y), 1 y 1, and 0, otherwise. 2 2 a. LetU Y. Find the pdf of U. b. Find E(U) using the pdf of U.