Stat Summer 2012 Exam 1. Your Name:

Transcription

1 Stat Summer 2012 Exam 1 Your Name: Your Section (circle one): Sveinn (08:40) Glen (09:50) Mike (11:00) Instructions: Show your work on ALL questions. Unsupported work will NOT receive full credit. Decimal answers should be exact, or to exactly 2 decimal places. (Examples: if it is.25 use.25, if it is say then use.0089.) You are responsible for upholding the Honor Code of Purdue University. This includes protecting your work from other students. Please write legibly. If a grader cannot read your writing, NO credit will be given. You are allowed the following aids: a one-page 8.5" x 11" handwritten (in your handwriting) cheat sheet, a scientific calculator, and pencils or pens. 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 of your questions at one time. You must show your student ID (upon request), turn in your cheat sheet and sign the class roster when you turn in your exam to your instructor. Turn off your cell phone before the exam begins. Question Points Possible Points Earned Cheat Sheet 1 Total 100

2 1. 3% of basketball cards are Hall of Fame cards. For each of the following scenarios, write the correct distribution and its parameter(s). If an approximation can be used, write both the exact and the approximate distributions and parameters to receive full credit. (15 points, 3 points each) a) Glen is buying one basketball card at a time. Let X denote the number of cards he buys until getting the first Hall of Fame card. b) John buys, on average, two basketball cards every month. Let Y denote the number of basketball cards he buys in Assume the number of cards he buys are independent from month to month. c) Sveinn has 120 basketball cards. Let X denote the number of Hall of Fame cards he owns. d) Vivian wants to have 5 Hall of Fame cards. Let Y denote the number of cards she needs to buy until getting the 5th Hall of Fame card. e) Mike bought a box of 200 cards in which there are 6 Hall of Fame cards. He randomly pick up 3 of them without replacement. Let X denote the number of Hall of Fame cards in this sample.

3 2. A group of 15 people are going to Six Flags. 6 of them are Caucasian, 5 Asian, and 4 of another race. 4 people are randomly selected for the new attraction X-Flight. Let X denote the number of Asians picked. (18 points) a) Identify the distribution of X and its parameter(s). (4 points) b) What is the probability that 3 Asians are picked? (3 points) c) What is the probability that 2 Asians and 2 Caucasians are picked? (3 points) d) What is the probability that at least one Asian and at least one Caucasian are picked? (4 points) e) Suppose next year there will be 300 people from the College of Science attending Six Flags, in which 130 are Caucasian, 110 Asian, and 60 of another race. Again, they randomly pick 4 people to try another new attraction, Mr. Freeze. What is the approximate probability that 2 Asians will be picked? (4 points)

4 3. Consider the following probability mass function (PMF) of random variable X: p x = x+4 20, x = 2, 1, 0, 1, 2 0, otherwise (14 points) a) Find the probability that X is less than 0.5 (3 points) b) Find E[X] (3 points) c) Find Var[2X-5] (3 points) d) Find E[(3X 2) 2 ] (5 points)

5 4. A special roulette board has 40 spaces: 20 red, 17 black, and 3 green. During each spin a ball lands on one of the 40 spaces, with each space having equal probability. One round consists of 10 spins of the roulette wheel. You win a round if the ball lands on green at least twice. (13 points) a) What is the probability of winning a round? (3 points) b) What is the probability the first round you win comes on the 4 th round you play? (3 points) c) What is the probability the first round you win comes before the 8th round you play? (3 points) d) Given your 3 rd win came on the 7 th round you play, in which round is your expected 8 th win? (4 points)

6 5. (10 points) a) A library has 800,000 books and the librarian wants to encode each by using a code word consisting of 3 uppercase letters followed by two numbers. Are there enough code words to encode all of these books with different code words? (3 points) b) If we put five math, six biology, and eight history books together on a bookshelf. What is the probability all the math books are together? Assume every books is distinct. (3 points) c) Assume we now have a collection of five math, six biology, and eight history books, and we want to sample five of them without replacement. How many different collections are possible if at least 2 Math books must be selected? (4 points)

7 6. Suppose Lionel Messi plays in 90% of the games for the Argentina National Football team. If Messi plays, Argentina has 0.6 probability of winning, 0.25 probability of drawing the match, and some probability of losing the match. If Messi does not play, Argentina has 0.5 probability of losing, 0.2 probability of drawing, an some probability of winning. (13 points) a) Draw an appropriate tree diagram to illustrate this problem. (3 points) b) What is the probability that Argentina wins the match? (3 points) c) Knowing that Argentina wins or draws the match, what is the probability Messi plays? (4 points) d) Is "Argentina wins the match" independent from "Messi plays"? Explain your reason. (3 points)

8 7. Andreana is looking forward to the London 2012 Summer Olympics. During each of the 19 competition days, she intends to invite all of her 120 Facebook friends over to her house. On any given day, she expects each friend to show up with probability 0.04, and they show up independently of one another. Let X denote the number of friends that show up on any given day. (17 points) a) State the exact and approximate distributions of X along with the parameters. State why the approximation is valid. (4 points) b) What is the approximate probability that between 3 and 5 friends, inclusive, show up on any given day? (3 points) c) What is the exact probability that at least 3 friends show up on any given day? (4 points) d) What is the probability that there will be more than 16 days where at least 3 friends show up? (3 points) e) What is the expected number of days that at least 3 friends show up? (3 points)