Concordia University Department of Mathematics & Statistics Course Number STAT 349/2 Examination Date Final Sections All Pages December 2013 4 Instructors A. Sen, W. Sun Special Instructions: . Show your work in sucient detail and clearly identify your answer. . Approved calculators only. . A one-page formula sheet is allowed. 1. An urn contains n red and m black balls. They are withdrawn from the urn, one at a time and without replacement. Define for i = 1, . . . , n, 8 < 1, if red ball i is taken before the second black ball is chosen, Xi = : 0, otherwise. Obtain E[X1 + + Xn ] and Var[X1 + + Xn ]. [14 marks] 2. Two players take turns shooting at a target, with each shot by player i hitting the target with probability pi , i = 1, 2. Shooting ends when the target has been hit twice. Let mi denote the mean number of shots needed for the first hit when player i shoots first, i = 1, 2, and 1 the mean number of shots taken until the shooting ends when player 1 shoots first. Find m1 , m2 and 1 . [12 marks] Stat 349/2 Final December 2013 Page 2 of 4 3. Let U1 , U2 , . . . , be independent uniform (0, 1) random variables, and define Nx by Nx = min{n 1 : U1 + + Un > x}, 0 x 1. Let f (x) = E(Nx ). (a) By conditioning on U1 = u show that f (x) satisfies the equation f (x) = 1 + Z x 0 f (u)du, f (0) = 1. (b) Obtain a dierential equation for f (x) by dierentiating both sides of the equation in part (a) and hence show that f (x) = ex , 0 x 1. [12 marks] 4. At all times, an urn contains 3 balls some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing heads is flipped. If heads appears, then a ball is chosen at random from the urn and is replaced by a white ball; if tails appears, then a ball is chosen at random from the urn and is replaced by a black ball. Let Xn denote the number of white balls in the urn after the n-th stage, n 0. Obtain the transition probability matrix of this Markov chain and find the proportion of time the chain spends in each state in the long run. [12 marks] 5. Three machines, 1, 2 and 3, have independent lifetimes. The amount of time that machine i functions is an exponential random variable with rate i, i = 1, 2, 3. Suppose that all machines are initially in use. (a) Find the probability that machine 3 is not the last one to fail. (b) Given that machine 1 fails first among the three machines, find the probability that machine 3 is the last one to fail. [10 marks] Stat 349/2 Final December 2013 Page 3 of 4 6. In a post oce run by two clerks, each customer only needs to be served by one of the clerks. The service times are independent exponential random variables with rates i , i = 1, 2. Suppose that two customers A and B are being served and customer C is on her way to the post oce. The amount of time it takes C to get to the post oce is exponentially distributed with rate . When C reaches the post oce, if clerk 1 is free then she starts her service with clerk 1 immediately; if clerk 1 is busy but clerk 2 is free then she starts her service with clerk 2 immediately; otherwise she has to wait and start her service with the first available clerk. Find the expected time for C to finish her service. [13 marks] 7. Buses arrive at a station according to a Poisson process with rate 2 per hour. Start from 0pm. (a) Find the probability that the second bus arrives after 1:30pm. (b) If there is no bus arriving between 1pm and 2pm, what is the expected number of buses arriving before 5pm? (c) If the waiting time T1 for the first bus is less than the inter-arrival time T2 between the first and second bus, what is the probability that the inter-arrival time T3 between the second and third bus is also less than T2 ? (d) If exactly two buses have arrived during the hour from 0pm to 1pm, find the distribution function of the arrival time of the second bus and hence its expectation. (e) Suppose that passengers arrive at the station according to another independent Poisson process with rate 6 per hour. All the passengers will wait until a bus arrives, and the bus takes away all the passengers Stat 349/2 Final December 2013 Page 4 of 4 waiting at the station. Find the probability that the second passenger is taken away by the second bus. [15 marks] 8. A job shop consists of two machines and two repairmen. The amount of time a machine works before breaking down is exponentially distributed with mean 12. Suppose the amount of time it takes a single repairman to fix a machine is exponentially distributed with mean 9. (a) If at time 0 both repairmen are busy, find the expected time until both of them are free. (b) Find the average number of machines in use. 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