410Hw06 - STAT 410 Summer 2009 Homework #6 (due Wednesday,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
STAT 410 Summer 2009 Homework #6 (due Wednesday, July 15, by 4:00 p.m.) 1. Let X be a continuous random variable with probability density function ( 29 β 1 X α β β α x e x x f - - = , x > 0, where α > 0, β > 0. ( X has a Weibull distribution. ) Consider Y = β X . What is the probability distribution of Y? 2. Let X have an exponential distribution with θ = 1; that is, the p.d.f. of X is f ( x ) = e x , 0 < x < . Let T be defined by T = ln X. a) Show that the p.d.f. of T is g ( t ) = e t e e t , < t < , which is the p.d.f. of an extreme value distribution. b) Let W be defined by T = α + β ln W, where < α < and β > 0. Show that W has a Weibull distribution. 3. 3.4.11 Let the random variable X have the p.d.f. f ( x ) = 2 2 2 2 x e - π , 0 < x < , zero elsewhere. Find the mean and the variance of X. Hint: Compute E ( X ) directly and E ( X 2 ) by comparing the integral with the integral representing the variance of a random variable that is N ( 0, 1 ).
Background image of page 1

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

View Full DocumentRight Arrow Icon
4. 3.4.10 If e 3 t + 8 t 2 is the mgf of the random variable X, find P ( 1 < X < 9 ). 5. 3.4.28 Let X 1 and X 2 be independent with normal distributions N ( 6, 1 ) and N ( 7, 1 ), respectively. Find P ( X 1 > X 2 ). Hint: Write P ( X 1 > X 2 ) = P ( X 1 – X 2 > 0 ) and determine the distribution of X 1 – X 2 . 6.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

410Hw06 - STAT 410 Summer 2009 Homework #6 (due Wednesday,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online