This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability & Statistics I Assignment 3 Due Date: October 22, 2010 (Hand in your solutions for Questions 2, 11, 13, 16, 19, 22, 26, 28) 1. Let X be a random variable with probability density function ( ) ( ) ⎩ ⎨ ⎧ < < − − = otherwise 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 . ( ) 7 . 3 . ≤ < − 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 ( ) ⎪ ⎩ ⎪ ⎨ ⎧ ≤ > = 10 10 10 2 x x x x f . (a) Find the cumulative distribution function of X . (b) Find . ( ) 15 ≥ 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 6 such types of devices at least 3 will function for at least 15 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 < < u . (a) Show that ( ) ( ) x F u x u F ≤ ⇔ ≤ − 1 for all ( ) 1 , ∈ 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 , , 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 10/11 4. A random variable Y is said to follow the double exponential distribution if it has the density function ( ) y e y f λ λ − = 2 1 , ∞ < < ∞ − y where > λ ....
View
Full
Document
This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.
 Spring '08
 SMSLee
 Statistics, Probability

Click to edit the document details