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  # NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION ST2131/MA2216 PROBABILITY
  
  ## (Semester 2: AY 20102011) April/May 2011 Time allowed: 2 hours
  
  INSTRUCTIONS TO CANDIDATES 1. This examination paper contains SIX (6) questions and comprises THIRTEEN (13) printed pages. 2. Answer ALL questions. Marks for each question are indicated. The total marks for this paper is 70. 3. Candidates may use any calculator. However, they should lay out systematically the various steps in the calculations. 4. This is a CLOSED BOOK examination. Candidates may bring in ONE (1) two-sided, A4 size help sheet. 5. Write down your matriculation number and seat number neatly in the boxes provided below. Answers should be written on this booklet. This booklet will be collected at the end of the examination.
  
  Matriculation Number
  
  Seat Number
  
  Question
  
  Total
  
  Score
  
  PAGE 2
  
  ST2131/MA2216
  
  Question 1 Players A and B play a sequence of independent games in which the two players take turn in rolling a fair die of 6 faces. The 6 faces are labeled as 1, 2, 3, 4, 5 and 6. Player A rolls the die once in game 1. Then, player A wins and the sequence stops if the die lands on 6, or otherwise they proceed to game 2. In game 2, the die is rolled for the second time, but by player B. Then, player B wins and the sequence stops if the die lands on either 6 or 5, or otherwise they proceed to game 3. In game 3, the die is rolled for the third time, but by player A. Then, player A wins and the sequence stops if the die lands on any one of 6, 5 or 4, or otherwise they proceed to game 4. The sequence continues in this manner until the end of game 6. (i) Find the probability that player A wins at game 5. [4 marks]
  
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  (ii) Suppose player A wins the sequence. Find the probability that player A wins at game 3. [4 marks]
  
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  Question 2 Suppose that a certain disease is present in 10% of the population in a region, and that there is a screening test designed to detect the presence of the disease in this population. Unfortunately, the test does not work perfectly; it is possible that the test shows a negative result for a person who has the disease, or it shows a positive result for a person who has no disease, or sometimes the result is indecisive (that is, neither positive nor negative). The following table shows the proportion of times that the test produces various results after diagnosing every person in this population. For instance, it shows that 2% of the population have the disease and the test showed a negative result. Positive (P) a b Test Result Indecisive (I ) 0.05 0.01 Negative (N ) 0.02 d [3 marks]
  
  Disease present (D) Disease absent (Dc ) (i) Find the value of a.
  
  (ii) Suppose the events N and D are independent, show that d = 0.18.
  
  [3 marks]
  
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  (iii) If a person selected at random from this population has no disease, what is the probability that the test shows a positive result? [3 marks]
  
  (iv) If the values of a, b and d in the table are all known, can you nd the total number of cases in which the test gives a correct result in detecting the disease in this population? In one short sentence, give a reason to support your answer. [3 marks]
  
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  Question 3 An urn contains N balls numbered 1 through N . A sample of n balls (where n < N ) are randomly selected. Let X and Y be the minimum and maximum of the chosen numbers respectively. (i) If the balls are selected without replacement, nd the probability mass function of Y . [3 marks]
  
  (ii) If the balls are selected with replacement, nd the probability mass function of Y . [3 marks]
  
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  (iii) If the balls are selected without replacement, nd P(Y X = 5) for N = 50, n = 5. Round the answer to 6 decimal accuracy. [4 marks]
  
  (iv) If the balls are selected with replacement, nd P(Y X = 1) for N = 50, n = 5. Round the answer to 6 decimal accuracy. [4 marks]
  
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  PAGE 8 Question 4 Let R be a random variable with the probability density function f (r) = 0.5642er ,
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  ST2131/MA2216
  
  < r < .
  
  (i) Let W be the absolute value of R. Find the probability density function of W . [3 marks]
  
  ## (ii) Let X = R2 . Find the probability density function of X .
  
  [3 marks]
  
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  (iii) Let Y be a random variable such that ln(Y ) has the same distribution as R. Find the probability density function of Y . [3 marks]
  
  (iv) Rank the 3 probabilities P(0 < W < u), P(0 < X < u) and P(0 < Y < u) from the smallest to the largest, where u is a constant such that 0 < u < 1. [7 marks]
  
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  PAGE 10 Question 5
  
  ST2131/MA2216
  
  (a) Bus A arrives at the Computer Center bus stop at a time that is uniformly distributed between 9:00 am and 10:00 am. Bus B arrives at the same bus stop at a time that is uniformly distributed between 9:00 am and the arrival time of bus A. If you have been waiting for bus B since 9:00 am, how long would you expect to wait? [6 marks]
  
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  (b) Let X and Y be two continuous random variables with nite expectations. The joint probability density function of X and Y , f (x, y), is dened such that f (x, y) = 0, for x > y and f (x, y) 0, for x y. Show that E (X ) E (Y ). [4 marks]
  
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  PAGE 12 Question 6
  
  ST2131/MA2216
  
  (a) The popular restaurant chain Hairy Jones has two branches in the cities of Tam-Pines and Bee-Sam. The probability density function of the weekly prot of each branch, in thousands of dollars, is given by f (x) = x/6, if 2 < x < 4, 0, otherwise.
  
  Assuming that the prot of each branch is independent of the other, what is the probability that one branch makes at least \$1000 more than the other in the coming week? [5 marks]
  
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  (b) Let X be a discrete random variable with probability mass function given by P(X = k) = 2 1 3
  k
  
  k = 1, 2, 3, . . .
  
  END OF PAPER
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