Stat 100A -Intro ProbabilityPractice Midterm 2 J. SanchezUCLA Department of StatisticsDO NOT DETACH ANY PAGES FROM THIS EXAM. EXAM MUST STAY STAPLED DURING THEWHOLE EXAM.UCLA ID: − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −− NO ID NO EXAM.SHOW AT BEGINNING OF EXAM and THROUGHOUT THE EXAM. ID MUST BE MARKED ONSCANTRON, ON EXAM AND ON BOTH SIDES OF THE CHEAT SHEET.LAST NAME (Please, PRINT): − − − − − − − − − − − − − − − − − − −− FIRST NAME: − − − − − − − − −−NAMA MUST BE ON BOTH SIDES OF CHEAT SHEET FROM BEGINNING OF EXAM, ON THE EXAMAND ON THE SCANTRON.Instructions• Closed books, closed notes.• Phones and other electronic devices must be disconnected before you enter the classroom and not turn onagain until you are out of the room. While in the classroom, they must be in your backpack and your backpackon the floor. Phones in pockets will lead to big loss of points in the exam.• Answer for multiple choice questions will be marked in scantron AND the exam. Work will not be read inmultiple choice;Failure to mark your name, ID or some answers will result in point deduction from the examgrade.• Left handed students will sit in a seat for left-handed students. The professor will tell students where to sit.Please, let the professor know that you are left handed ahead of time.• ID must be ready to show BEFORE and at all times during the exam. NO ID, no exam.• This midterm must show your individual work. Talking to others during the midterm, not adhering to theabove, sharing information or breaking any other aspect of the student code of conduct at UCLA will not betolerated. You can not exchange papers or information. All your things must be on the floor. You may notuse the empty seats next to you to put things. Close the tables. Honor code applies. You may not talk or useyour electronic devices from the moment the professor starts distributing the midterm until they have beencollected from everybody.• Only scientific calculator and two sides of a 8 × 11 cheat sheet, and pen or pencil and exam and scantron canbe on your desk. Everything else (disconnected) in your backpack and your backpack on the floor. Close thetables of the chair near you.• Cheat sheet can have only formulas and definitions, no solved problems, no examples of any kind, no proofs,no numerical examples, no intermediate steps and no drawings or graphs of any kind.YOUR NAME MUSTBE ON CHEAT SHEET AT ALL TIMES. Be ready to show your cheat sheet when the instructor requests it.February 17, 2014 1Stat 100A -Intro ProbabilityPractice Midterm 2 J. SanchezUCLA Department of StatisticsThe cheat sheet must be written all in English. Cheat sheets that do not comply will result in lower grade inthe exam.• In questions where you show work, you must use the same notation we have used in lecture and the textbook.• Phones and all other electronic devices must be disconnected and in your backpack. Your backpack must beon the floor.• Failure to follow instructions given here will result in loss of points in the exam in a first warning. Honorcode applies. Familiarize yourself with student code of conduct by visiting the links provided in the coursesyllabus.The exam contains• Cover page (write your name and ID)• Midterm will have Multiple choice questions. Answers must be marked on exam and the scantron with yourname and ID. Work is not required and will not be read.• Question where you must show work. Answer must be written on the exam.MULTIPLE CHOICE QUESTIONS. ONLY ONE ANSWER IS CORRECT. CHOICE MUST BE MARKEDON THE SCANTRON, AND ALSO HERE ON THE EXAM. ONLY THE SCANTRON WILL BE GRADED.NO MARKS ON SCANTRON OR MORE THAN ONE MARK WILL RESULT IN 0 POINTS FOR THEQUESTION NOT MARKED, EVEN IF IT IS MARKED ON THE EXAM. Work near the multiple choicequestion will not be read.Problem 1. A random variable X has cumulative distribution functionF(x) =126(2x2+ x − 10) 2 ≤ x ≤ 4The probability that X is between 2.8 and 4 is(a) 0.4231(b) 0.2291(c) 0.0023(d) 0.0384(e) 0.6739Solution 1. F(4)= 1.1 − (1/26) ∗ (2 ∗ (2.82) + 2.8 − 10) = 1 − 0.3261 = 0.6739February 17, 2014 2Stat 100A -Intro ProbabilityPractice Midterm 2 J. SanchezUCLA Department of StatisticsProblem 2. What is the constant k that makes the following function a valid density function?f (x) = kx9(1 − x)20 ≤ x ≤ 1(a) 0.00151(b) 2(c) 210(d) 660.06(e) 12Solution 2.R10kx9(1 − x)2dx = 1 by definition of density of a continuous random variable.This gives 1/k = 0.0015; k = 660Problem 3. A continuous random variable has the following probability density functionf (x) =x20 ≤ x ≤ 2The 50th percentile is(a) 1.0(b) 5/16(c) 0.034(d) 0.5(e) 1.414Solution 3. The 50th percentile is the median.Rc0x2dx = 0.5. Solve for c=1.414Problem 4. The density of a continuous random variable X is such that the following applies: P(X < 16) =0.75, P(X < 10) = 0.25, P(X < 12) = 0.5, P(X < 19) = 0.9, P(X < 5) = 0.10. Which of the following is true:(a) The Interquartile range is 6, and the distribution is symmetric(b) The Interquartile range is 4, and the distribution is skewed(c) The interquartile range is 6 and the distribution is skewed.(d) The interquartile range is 7, and the distribution is skewed.(e) The median is 19February 17, 2014 3Stat 100A -Intro ProbabilityPractice Midterm 2 J. SanchezUCLA Department of StatisticsSolution 4. The interquartile range is 6 and the distribution is skewed.Problem 5. The distribution of X, the time (in minutes) it takes women between 50 and 55 to run a 10k race, is suchthat the event A (X between 40 and 60 minutes) has P(40 < X < 60) = 0.8; the event B, taking between 50 and 90minutes has P(50 < X < 90) = 0.2; and the event C, taking between 50 and 60 minutes has P(50 < X < 60) = 0.1.For a runner woman chosen at random from the population in this age group, compute the probability that she is ineither A or B (A ∪ B).(a) 0.9(b) 0.3(c) 0.8(d) 0.2(e) 0.54Solution 5. P(A ∪ B) = P(a) + P(B) − P(AB) = 0.9Problem 6. The service times X1, X2, ....., X10, of 10 teller windows in a bank are independent and identicallydistributed random variables. Each Xiis exponential with an expected value of 3.2 minutes. In a given day, 10independent customers arrive at 3:00 pm and each is serviced by a different window.What is the joint probability that all 10 customers will be there at
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