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1. (a) W = log (1). Here log is log base e. i. Find the range of W. ii. Find the density of W. The joint distribution of two random variable X and Y is (b) Part (a) and (b) of this question are independent. Suppose a random variable U follows Uniform(0,1) distribution. Define fxy(x, y) = 8 xy 1(0

1. (a) W = log (1). Here log is log base e. i. Find the range of W. ii. Find the density of W. The joint distribution of two random variable X and Y is (b) Part (a) and (b) of this question are independent. Suppose a random variable U follows Uniform(0,1) distribution. Define fxy(x, y) = 8 xy 1(0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. Part (a) and (b) of this question are independent. (a) Suppose a random variable \( U \) follows Uniform(0,1) distribution. Define \[ W = \log \left( \frac{U}{1-U} \right). \] Here log is log base \( e \). i. Find the range of \( W \). ii. Find the density of \( W \). (b) The joint distribution of two random variables \( X \) and \( Y \) is \[ f_{X,Y}(x,y) = 8xy \, 1_{(0<y<x<1)}. \] Here \( 8xy \, 1_{(0<y<x<1)} \) means \( 8xy \) when \( 0 < y < x < 1 \) and zero otherwise. Define \( Z = \frac{Y}{X} \). i. Find the range of \( X \). Find the range of \( Z \). ii. Find the joint density \( g_{X,Z}(x,z) \) of \( X \) and \( Z \). Hint: Use the change of variable formula for \( (X,Y) \) to \( (X,Z(X,Y)) \) in Lecture 31. iii. Find the marginal density of \( X \). iv. Find the marginal density of \( Z \). v. Are \( X \) and \( Z \) independent?
Transcribed Image Text:1. Part (a) and (b) of this question are independent. (a) Suppose a random variable \( U \) follows Uniform(0,1) distribution. Define \[ W = \log \left( \frac{U}{1-U} \right). \] Here log is log base \( e \). i. Find the range of \( W \). ii. Find the density of \( W \). (b) The joint distribution of two random variables \( X \) and \( Y \) is \[ f_{X,Y}(x,y) = 8xy \, 1_{(0<y<x<1)}. \] Here \( 8xy \, 1_{(0<y<x<1)} \) means \( 8xy \) when \( 0 < y < x < 1 \) and zero otherwise. Define \( Z = \frac{Y}{X} \). i. Find the range of \( X \). Find the range of \( Z \). ii. Find the joint density \( g_{X,Z}(x,z) \) of \( X \) and \( Z \). Hint: Use the change of variable formula for \( (X,Y) \) to \( (X,Z(X,Y)) \) in Lecture 31. iii. Find the marginal density of \( X \). iv. Find the marginal density of \( Z \). v. Are \( X \) and \( Z \) independent?
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