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assignment 3

assignment 3 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 Probability and Statistics: Foundations of Actuarial Science Assignment 3 Due Date: October 25, 2010 (Hand in your solutions for Questions 3, 7, 17, 23, 37, 41, 46, 51) 1. Trains headed for destination A arrive at the train station at 15-minute intervals starting at 7 A.M., whereas trains headed for destination B arrive at 15-minute intervals starting at 7:05 A.M. (a) If a certain passenger arrives at the station at a time uniformly distributed between 7 and 8 A.M. and then gets on the first train that arrives, what proportion of time does he or she go to destination A ? (b)What if the passenger arrives at a time uniformly distributed between 7:10 and 8:10 A.M.? 2. Let U denote a random variable uniformly distributed over ( ) 1 , 0 . Compute the conditional distribution of U given that (a) ; a U > (b) a U < where . 1 0 < < a 3. ( Random number generation ) A general method for simulating a random variable—called the inverse transformation method —is based on the following function: ( ) ( ) { } : min 1 u x F x u F = where F is a distribution function and 1 0 < < u . (a) Show that ( ) ( ) x F u x u F 1 for all ( ) 1 , 0 u and real x . (b) Use the result in (a), or otherwise, to show that if F is a distribution function and U is a uniform random variable from ( ) 1 , 0 , then ( ) U F X 1 = will be a random variable with distribution function F . (c) Write down the procedure to generate a random variable from () λ Exp . 4. There are 100 enrolled students in a statistics class. Example sheets are required for a tutorial, and the economy-conscious tutor, anticipating a certain amount of absenteeism, plans to produce the minimum number of sheets required to ensure a probability of 0.95 that there will be enough to go round those students who turn up. He assumes that each student has independent probability 0.07 of being absent. Find the number of sheets he should produce. 5. A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times, and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct. What is the probability that he would have done at least this well if he had no ESP? P. 1

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10/11 6. Suppose that airplane engines will fail, when in flight, with probability independently from engine to engine. If an airplane needs a majority of its engines operative to make a successful flight, for what values of p is a 5-engine plane preferable to a 3-engine one? p 1 7. The price of a particular highly volatile stock either increases 25% or decreases 20% in any given week. The probability of an increase in any week is 55%, independent of the stock’s performance on other weeks. The current price of the stock is \$10. (a) Determine the probability that the stock’s price exceeds \$15 four weeks from now.
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assignment 3 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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