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Unformatted text preview: 1 ISyE 3044 Old Questions from Exam #1 1. Consider the integral = integraltext 1 / 2 sin( t ) dt . (a) Use the following 10 pseudo-random numbers to compute an estimate of I . 0.60 0.73 0.35 0.08 0.99 0.47 0.22 0.16 0.54 0.87 (b) Compute an approximate 90% confidence interval for . 2. Short questions. (a) Suppose that U is a Uniform(0 , 1) random variable. What is the distribution of X =- 1 5 ln U ? Find the mean of X . (b) The discrete random variable X has the following probability function: k 1 2 3 4 5 P ( X = k ) 0.30 0.35 0.20 0.10 0.05 Use the uniform random number U = 0 . 79 to generate an observation for X . 3. Short questions. Customers arrive at a post office branch according to a Poisson process with a rate of 2 per minute. (a) What is the expected number of arrivals between 10:30 and 10:40 a.m.? (b) Assume that the post office opens at 9 a.m. with no customers present. What is the probability that the third customer will arrive after 9:02 a.m.? (c) Suppose that no customer has arrived between 10 and 10:05 a.m. What is the probability that the next customer will arrive within the next minute? 4. Parts are machined on a drill press. They arrive at a rate of one every 5 3 minutes (uniformly distributed) and it takes 3 2 minutes to machine them. The machine under- goes preventive maintenance: The times between maintenance are equal to 55 minutes and maintenance lasts 5 minutes. We developed a simulation model and performed 10 independent replications, each for 8 hours. The objective is the computation of a 90% confidence interval for the mean time in system (cycle time) of a part. (a) The following table shows selected part of the output from the 10 runs. Use the information to compute the 90% confidence interval for the mean cycle time. (b) Compute a 90% confidence interval for the mean time in the drill press queue. (Hint: The cycle time is the sum of the queue time and the process time.) 2 Table 1: Selected Simulation Output Run Average Cycle Time 1 3.94 2 4.15 3 3.52 4 3.86 5 3.92 6 3.81 7 3.91 8 3.51 9 3.41 10 3.94 5. Short questions. (a) Let X and Y be i.i.d. exponential( = 1) random variables. Find P ( X + Y 2). (b) Suppose X 1 ,... ,X 20 are i.i.d. exponential( = 1 / 2). Use the central limit theorem to approximate P ( 20 i =1 X i 50). (c) Suppose X Poisson(1) and Y Poisson(2) are independent. Find P ( X + Y 3). (d) The Atlanta Braves are currently winning 60% of their games. They play 5 games during a week. What is the probability that they will win more games than they lose? (e) The time to failure for a computer chip is known to have the Weibull distribution with parameters = 1 / 2 and = 1 / 300. Find the fraction of chips that survive 600 days....
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- Spring '08