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hw4-16.pdf

# hw4-16.pdf - 15/16 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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15/16 P. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801/2901 Probability and Statistics: Foundations of Actuarial Science Assignment 4 Due Date: April 29, 2016 Important notice. (i) Hand in your solutions for Questions 4,15,19,22,26,34,43,44 (ii) For review, consider Questions 5, 11, 17, 21, 27, 37, 42, 46. If you need help for these questions, please contact your tutor and lecturer during their office hours. (iii) You may forget all the other exercises. There are listed here for some convenience only. 1. Let X be a random variable with pdf given by   4 1 3 x x f , x 0 . Let X Y log . Find the pdf of Y . 2. Let U be a random variable uniformly distributed in 1 , 0 . The random variable U U b a X 1 log is said to follow the logistic distribution , where a and 0 b . Find the pdf of X . 3. Let   1 ~ l Exponentia X , and define Y to be the integer part of 1 X . (a) Find the pmf of Y . What well-known distribution does Y have? (b) Find the conditional distribution of 4 X given 5 Y . 4. Suppose 1 , 0 ~ N Z . (a) Find the pdf of 2 Z X . (b) Find the pdf of Z Y . (c) find the pdf of Z e W . (d) Find the mean and median of the random variable W defined in part (c). 5. Suppose 2 1 ~ X . Find the pdf of X Y .

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15/16 P. 2 6. A standard Cauchy random variable X has density function   2 1 1 x x f , x . (a) Determine X E . (b) Show that X Y 1 is also a standard Cauchy random variable. 7. Let X be a random variable distributed as exponential with parameter . Define a new random variable Y according to the following procedure: A fair coin is tossed once. Put X Y if a head turns up and X Y if otherwise. Find the probability density function of Y . 8. The random variables 1 Y and 2 Y are independently distributed, both with density   otherwise 0 1 if 1 2 y y y f . Let 2 1 1 1 Y Y Y U and 2 1 2 Y Y U . (a) Find the joint density of 1 U and 2 U . (b) Sketch the region where 0 , 2 1 , 2 1 u u f U U . (c) Find the marginal density of 1 U . (d) Are 1 U and 2 U independent? Why or why not? 9. Let X and Y be random variables with joint pdf otherwise 0 0 if 2 , 2 y x y xe y x f y . Let X Y W , X Y Z . Find the joint pdf of W and Z . 10. Suppose X and Y have joint density function otherwise 0 1 , 1 if 1 , 2 2 y x y x y x f . (a) Find the joint density function of XY U , Y X V . (b) Find the marginal pdfs of U and V . 11. Let X and Y be two random variables with joint probability density function ( pdf ) otherwise 0 0 for 9 , 3 x y e y x f x .
15/16 P. 3 Let Y X W , Y X Z . Determine the joint pdf of W and Z .

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