assignment 4

assignment 4 - 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 4 Due Date: November 18, 2010 (Hand in your solutions for Questions 4, 7, 12, 16, 19, 27, 28, 30) 1. Let X be a random variable with pdf given by () 4 1 3 x x f + = , < < x 0 . Let . Find the pdf of Y . X Y log = 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 . Find the pdf of X . 0 > b 3. Let , and define Y to be the integer part of 1 ~ Exp X 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 . Find the pdf of ( 1 , 0 ~ N Z ) (a) 2 1 Z X = ; (b) Z Y = . 5. The random variables and are independently distributed, both with density 1 Y 2 Y < < = 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 and . 1 U 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 and independent? Why or why not? 1 U 2 U P. 1
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10/11 6. 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 . 7. 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 . 8. Let X and Y be two continuous random variables having the joint pdf < + < < < < = otherwise 0 1 , 1 0 , 1 0 if 24 , y x y x xy y x f . Find the joint pdf of Y X Z + = and X W = . 9. Suppose X , Y , and Z are independently and identically distributed as . Derive the joint distribution of , 1 Exp Y X U + = Z X V + = , Z Y W + = . 10. The joint density function of X and Y is given by ( ) > > = + otherwise 0 0 , 0 if , 1 y x xe y x f y x . (a) Find the conditional density of X ,given y Y = , and that of Y , given . x X = (b) Find the density function of XY Z = . 11. Suppose . Find the cumulative distribution function of ( 1 , 0 ~ , U Y X iid ) Y X Z + = . 12. If X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter 1 = λ , find the distribution of (a) Y X Z + = and (b) Y X Z = . Assume independence. 13. Suppose and are independent exponential random variables with respective parameters 1 X 2 X 1 and 2 . (a) Find the distribution of 2 1 X X Z = . (b) Compute 2 1 X X P < . P. 2
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10/11 14. When a current I (measured in amperes) flows through a resistance R (measured in ohms), the power generated is given by (measured in watts). Suppose that I and R are independent random variables with densities R I W 2 = () < < = otherwise 0 1 0 if 1 6 x x x x f I , .
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assignment 4 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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