stat1301hw3

stat1301hw3 - 11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE Assignment 3 Due Date: October 26, 2011 (Hand in your solutions for Questions 2, 5, 10, 19, 21, 24, 26, 28, 34) 1. Let X be a random variable with probability density function () ( ) < < = otherwise 0 1 1 if 1 2 x x c x f . (a) What is the value of c ? (b) What is the cumulative distribution function of X ? (c) Find and . () X E () X Var (d) Evaluate . () 3 . 0 7 . 0 < < X P (e) Find the probability density function of 2 X Y = . 2. The probability density function of X , the lifetime of a certain type of electronic device (measured in hours), is given by () > = 5 0 5 5 2 x x x x f . (a) Find the cumulative distribution function of X . (b) Find . () 12 X P (c) Find and . () X E () X Var (d) Find ( ) X E . (e) Determine the lower quartile, median, and upper quartile of X . (f) What is the probability that of 8 such types of devices at least 3 will function for at least 12 hours? What assumptions are you making? 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 . P. 1

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11/12 4. Let X be a random variable distributed with the following pdf: () ( ) < < = otherwise 0 2 2 if 4 2 x x c x f . (a) Find the value of c . (b) Find the distribution function ( ) x F of X . (c) Show that () ( ) 1 = + x F x F for all x . What does it mean? (d) Determine a positive constant a that satisfies () 16 11 a a X P = . (e) Does there exists a positive constant a that satisfies () 16 3 a a X P = ? If so, what is it? 5. A random variable Y is said to follow the double exponential distribution if it has the density function () y e y f λ = 2 1 , < < y where 0 > . (a) Find the distribution function of
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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stat1301hw3 - 11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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