STAT 225 – Fall 2014 EXAM 2 NAME _____________________________ Your section (circle one): Patrick (7:30 am) Patrick (8:30 am) Pan (9:30 am) Pan (10:30 am) Courtney (7:30 am) Nathan (1:30 pm) Eric (4:30 pm) Alexa (8:30 am) Ce-Ce (2:30 pm) Kalyani (11:30 am) Ce-Ce (ONLINE) Kalyani (12:30 pm) Jiasen (3:30 pm) DIRECTIONS: IMPORTANT--- check both sides of all pages!!! 1. Show your work on questions 7-11. Unsupported work will NOT receive full credit. INCLUDE a probability statement on all questions that ask for you to determine a probability 2. Decimal answers should be exact or to exactly 4 significant digits. 3. You are responsible for upholding the Honor Code of Purdue University. This includes protecting your work from other students. 4. Please write legibly. If a grader cannot read your writing, NO credit will be given. 5. You are allowed the following aids: a) A one-page 8.5” x 11” HANDWRITTEN cheat sheet b) Scientific (non-graphing) calculator (in accordance with the syllabus) c) Pencils, pens, erasers 6. Cheat sheets with photocopied or printed information are not allowed and are subject to disciplinary action. 7. Instructors will not interpret questions for you. If you do have questions, wait until you have looked over the whole exam so that you can ask all your questions at one time. 8. TURN OFF your cell phone and put it away. Cell phones may NOT be used as a calculator! 9. Sign the class roster when turning in your exam. Present your student ID upon request. Question Points Possible 1-5 16 6 12 7 10 8 16 9 14 10 16 11 15 Cheat Sheet 1 TOTAL 100 Points Earned 1 For 1-5, Please write the letter of your answer on the line provided. _B_ 1. Let A be a random variable representing the time between arrivals of flights at Indianapolis International Airport. Then A is a/an ______________ random variable (3 points) A. B. C. D. E. Independent Continuous Countable Discrete Poisson _B_ 2. For a _________________ random variable, X, with n = 120 and 𝑝 = 1 250 . An appropriate approximation would be ____________________ . (The answers below fill in the blanks in the order given.) (3 points) A. B. C. D. E. Binomial; X*~Hypergeometric (N = 250, n = 150, p = 0.004) Binomial; X*~Poisson ( 𝜆 = 0.48) . Poisson; X*~ Binomial (n = 120, p = 0.48) Geometric; X*~Poisson ( 𝜆 = 0.48) Hypergeometric; X*~ Binomial( n= 120, p = 0.48) _D_ 3. Please select the scenario(s) in which the memoryless property can be applied. (3 points) A. B. C. D. E. X ~ Poisson(λ=1), 𝑃(𝑋 > 5 | 𝑋 > 3) X ~ Exponential(λ=1), 𝑃(𝑋 < 5 | 𝑋 < 3) X ~ Exponential( λ=1), 𝑃(𝑋 > 5 | 𝑋 < 3) X ~ Geo(p =0.01), 𝑃(𝑋 < 5 | 𝑋 > 3) Both C and D _E_ 4. Choose the statement below that is true. (3 points) A. B. C. D. E. The CDF of a continuous random variable may be decreasing. 𝐹𝑋 (𝑎) = 𝑃(𝑋 > 𝑎) For a continuous random variable, X, 𝑓𝑋 (𝑥) may be less than 0 The expected value of a discrete random variable is always an integer. The mean of a Poisson random variable equals its variance. _C_ 5. Let j be the 75th percentile of a Uniform(a=1, b=5) random variable. Let k be the standard deviation of an Exponential (λ = 0.2) random variable. Let l be the variance of a Poisson (λ = 3) random variable. Which of the following statements is true regarding the relationship between j, k, and l. (4 points) j = 4, k = 5, l = 3 so l < j < k A. 𝑗 < 𝑘 < 𝑙 B. 𝑘 < 𝑗 < 𝑙 C. 𝑙 < 𝑗 < 𝑘 D. 𝑗 = 𝑘 = 𝑙 E. Cannot determine without more information. 2 6. For each scenario below, state the distribution and parameter(s) of the random variable. ALSO write out the support (3 points each) a) Joe’s old car gets 3 flat tires per year on average. Let F be the number of flat tires that Joe’s car gets in 5 years. F~Poisson(λ = 3*5 = 15) support for F is 0, 1, 2, …. b) At Happy Hollow School, a recent poll of students showed that the favorite flavor of ice cream for 35% of the kids is vanilla. A certain 4th grade teacher has a class of 25 students. Let I be the number of students in this teacher’s class whose favorite ice cream is vanilla. I ~ Binomial(n=25, p = 0.35) support for I is 0, 1, 2, …., 24, 25 c) Customers arrive at a gas station at a rate of 2 every 5 minutes. Let T be the amount of time (in minutes) until the next customer arrives 2 5 T~Exponential(λ = 5) or T~Exponential(µ = 2 = 2.5) support for T is T > 0 d) There are 30 Skittles in a fun-sized bag of the candies. 17 of the skittles are red and the rest are other colors. An individual chooses 15 skittles from the bag without looking. Let R be the number of red skittles that the individual chose. R~Hypergeometric (N= 30, n=15, M = 17) or use p = 17⁄30 Support for R is 2, 3, 4, …, 14, 15 3 7. Let the random variable 𝑋 have the probability density function: 𝑎 𝑓𝑋 (𝑥) = 𝑒 𝑎𝑥 2 {𝑎 −𝑎𝑥 𝑒 2 𝑓𝑜𝑟 𝑥 < 0 𝑓𝑜𝑟 𝑥 ≥ 0 a) For any 𝑎 > 0, is 𝑓𝑋 a valid pdf? Why or why not? (Hint: 𝑒 𝑐 ≥ 0 for any 𝑐 and 𝑒 −∞ = 0) (3 points) 𝑎 −𝑎𝑥 𝑒 2 𝑎 > 0 for all 𝑥 ≥ 0, 𝑎 > 0. Similarly, 2 𝑒 𝑎𝑥 > 0 for all 𝑥 < 0, 𝑎 > 0. Additionally, ∞ 0 ∞𝑎 𝑎 ∫−∞ 𝑓𝑋 (𝑥)𝑑𝑦 = ∫−∞ 2 𝑒 𝑎𝑥 𝑑𝑥 + ∫0 2 𝑒 −𝑎𝑥 𝑑𝑥 = 1 2 1 + 2 = 1. Hence a valid pdf. b) Let 𝑎 = 9, derive an expression for the CDF 𝐹𝑋 (𝑥). (4 points) 𝑥 9 9 1 ∫ 𝑒 9𝑥 𝑑𝑥 = ( ) ( ) (𝑒 9𝑥 − 0) = 2 2 9 𝐹𝑋 (𝑥) = 0 −∞ 𝒆𝟗𝒙 , 𝟐 𝑓𝑜𝑟 𝑥 < 0 𝑥 9 9 1 9 −1 𝒆−𝟗𝒙 ∫ 𝑒 9𝑥 𝑑𝑥 + ∫ 𝑒 −9𝑥 𝑑𝑥 = + ( ) ( ) (𝑒 −9𝑥 − 1) = 𝟏 − , 𝑓𝑜𝑟 𝑥 ≥ 0. 2 2 2 2 9 𝟐 {−∞ 0 c) What is the probability that X is less than -0.2, when 𝑎 = 9? (3 points) 𝑃(𝑋 ≤ −0.2) = 𝐹𝑋 (−0.2) = 𝑒 −9∗0.2 2 = 𝟎. 𝟎𝟖𝟐𝟔 Or integrate the pdf from -∞ to -0.2 −0.2 9 ∫−∞ 2 𝑒 9𝑥 𝑑𝑥 = (29) (19) (𝑒 9𝑥 )| −0.2 = −∞ 𝑒 −9∗0.2 2 – 0 = 0.0826 4 8. Tornado intensity is measured on the Fujita scale. As the value of the Fujita scale increases from, F0, to F5, so does the intensity of the storm. In Indiana, tornados of F3 or higher intensity occur on average once every ten years. Let T be the time (in years) between tornadoes of this intensity in Indiana. Assume tornadoes of this intensity occur independently. a) What is the distribution and parameter(s) of T? What is the support for T? (3 points) 1 T~Exponential( λ = 10) or T~Exponential( µ = 10); support for T is T > 0 b) Given that there has not been a tornado of F3 or higher intensity in the last 6 years in Indiana, what is the probability that it will take between 11 and 15 years (total time) for the next tornado of that intensity to strike in Indiana? (4 points) P(11 < T < 15| T > 6) = P(5 < T < 9) by memoryless property 9 5 5 9 = F(9) – F(5) = 1 − 𝑒 −10 − (1 − 𝑒 −10 ) = 𝑒 −10 − 𝑒 −10 = 0.19996 c) What is the probability that 2 tornadoes of F3 or higher occur in Indiana from 2015 to 2025 and 5 tornadoes of F3 or higher occur in Indiana from 2037 to 2077? What distribution(s) and parameter(s) are you using? (5 points) Let X1 be the number of F3 or higher tornadoes from 2015 to 2025; X1~Poisson(λ = 1 10 1 ∗ 10 = 1) Let X2 be the number of F3 or higher tornadoes from 2037 to 2077; X2~Poisson(λ = 10*40 = 4) Since the time periods do not overlap---these are independent. P(X1 = 2 ∩ X2 = 2) = P(X1 = 2)*P(X2 = 2) = 𝑒 −1 ∗12 2! ∗ 𝑒 −4 ∗45 5! = 0.1839397 * 0.15629345 = 0.0287 d) Suppose that 6 tornadoes of F3 or higher occurred in Indiana between 1920 and 1960, but unfortunately the detailed records do not show exactly when these 6 tornadoes were. What is the probability that 2 of them occurred in the first 10 years of this time period (i.e. between 1920 and 1930? (4 points) Let Y be the number of these 6 tornadoes occurring in the first 10 years. Y~Binomial(n=6, p=.25) P(Y = 2) = (62)(0.25)2 ∗ (0.75)4 = 0.2966 5 9. An oil company conducts a geological study that indicates that an oil well should have a 20% chance of striking oil. (Notes on terminology: striking oil means that the well does produce oil; drilling for oil is the process of digging into the ground with machines to locate oil). Assume that each oil well is independent of the others. a) Let W be the number of wells drilled until oil is found. What is the distribution and parameter(s) of W? What is the support for W? (3 points) W~Geometric(p = 0.20) support is 1, 2, 3, ….. b) What is the probability that the company must drill at least 3 wells before finding the first well to strike oil? (3 points) P(W > 3) = P(W > 2) = (1-0.20)2 = 0.64 use Tail Probability Or P(W > 3) = 1 – P(W < 2) = 1 - (𝑃(𝑊 = 1) + 𝑃(𝑊 = 2)) = 1 − ( (0.2)1 + (0.2)1*(0.8)1) = 0.64 c) What is the probability that the third well to strike oil comes on the seventh well drilled? distribution and parameters are you using? (5 points) What Let J be the number of wells drilled until 3rd one to strike oil. J~Negative Binomial(r = 3, p = 0.2) P(J=7) = (62)(0.2)3 ∗ (0.8)4 = 15 ∗ 0.008 ∗ 0.4096 = 0.049152 d) It costs the oil company $175,000 to drill a well. What is the expected cost for the oil company to find 3 producing oil wells? (3 points) 3 Find E[175000*J] = 175000*E[J] = 175000*0.2 = $2,625, 000 6 10. Alice and Bob play a game. Each time the game is played, Alice flips a fair coin twice, while Bob draws two balls (without replacement) from an urn containing 4 black balls and 6 white balls. Alice wins the game if the number of heads she gets is greater than the number of black balls Bob gets. (Ties go to Bob.) Assume that Alice and Bob’s scores are independent. a) Let A denote the number of heads Alice gets. Let B denote the number of black balls Bob gets. State the distributions and parameter(s) of A and B. (4 points) A~Binomial (n = 2, p=0.5) B~Hypergeometric(N = 10, n = 2, M = 4) or use p=0.4 b) What is the probability that Alice gets at least one head? (3 points) P(A> 1) = 1 – P(A=0) = 1 − (20)(0.5)0 ∗ (1 − 0.5)2 = 0.75 c) What is the probability that Alice wins the game? (5 points) Alice wins if A = 1, and B = 0. Also Alice wins if A = 2 and B = 0 or 1 So P(Alice wins) = P(A=1 ∩ B=0) + P(A=2 ∩ B=0) + P(A=2 ∩ B=1) = P(A=1)*P(B=0) + P(A=2)*P(B=0) + P(A=2)*P(B=1) since Alice & Bob are independent = (21)(0.5)1 ∗ (1 − 0.5)1 ∗ 15 15 6 (4 0)∗(2) (10 2) + (22)(0.5)2 ∗ 6 (4 0)∗(2) (10 2) + (22)(0.5)2 ∗ 6 (4 1)∗(1) (10 2) = 24 = 0.5*45 + 0.25*45 + 0.25*45 = 0.3833 d) What is the probability that Bob wins the game? And on average, how times many must the game be played until Bob's first win? (4 points) P(Bob wins) = 1- P(Alice wins) = 1 -0.3833 = 0.6167 Let G be the number of games until Bob’s first win. So G~Geometric(p=0.6167) 1 E[G] = 0.6167 = 1.6215 7 11. Nicole always arrives at her bus stop at 10:05 am, knowing that the arrival of the bus varies evenly from 10:05 am to 10:20 am. Let X be the amount of time (in minutes) that Nicole waits for the bus to arrive. a) What is the probability that Nicole will have to wait longer than 8 minutes? What distribution and parameters are you using? (4 points) X~Uniform(a=0, b=15) 8−0 P(X > 8) = 1 - 15−0 = 0.4667 b) What is the probability the bus will come between 10:12 am and 10:18 am? (3 points) 10:12  7 minutes and 10:18 13 minutes 13−0 P(7 < X < 13) = 15−0 − c) 7−0 6 = = 0.4 15−0 15 Nicole has been keeping a record of her wait times for the bus, what is the 40th percentile of her wait times? What time is that on the clock? (4 points) 𝑎−0 Want F(a) = 0.40  15−0 = 0.4 solve for a = 6 minutes On the clock that is 10:05 + 6 minutes = 10:11 am d) After Nicole gets on the bus, she has a 10 minute ride and then a 4 minute walk to her class. Her class begins at 10:30 am. What is the probability that she will be on time for class? (4 points) So 10 minute ride and 4 minute walk means that she must get on the bus by 10:16 am or she will be late. 10:16 am is 11 minutes waiting time. 11−0 Find P(X < 11) = 15−0 = 0.7333 8