assignment 4

# assignment 4 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 , .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 7

assignment 4 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online