Answered: Is x and y independent?

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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Question

Is x and y independent?

Expert Solution

Step 1: To test the independence of 2 random variables.

Given ,the joint probability density function (pdf) of (X,Y) is

f(x,y)= $2 over 7 left parenthesis x plus 2 y right parenthesis space comma space space 0 less than x less than 1 comma space space 1 less than y less than 2$

Marginal pdf of X is

$f subscript 1 left parenthesis x right parenthesis equals integral subscript 1 superscript 2 2 over 7 left parenthesis x plus 2 y right parenthesis space d y$ = $2 over 7 left square bracket x y equals 2 y squared over 2 right square bracket space a t space y equals 1 space a n d space 2$

= $2 over 7 left square bracket x left parenthesis 2 minus 1 right parenthesis space plus left parenthesis 4 minus 1 right parenthesis right square bracket space equals 2 over 7 left square bracket x plus 3 right square bracket$ , 0<x<1

Marginal pdf of Y is $f subscript 2 left parenthesis y right parenthesis space space equals integral subscript 0 superscript 1 2 over 7 left parenthesis x plus 2 y right parenthesis d x space equals 2 over 7 left square bracket x squared over 2 plus 2 x y right square bracket space a t space x equals 0 space a n d space 1$

= $2 over 7 left square bracket 1 half plus 2 y right square bracket space equals 1 over 7 left parenthesis 1 plus 4 y right parenthesis$ , 1<y<2

Product of two marginals

$f subscript 1 left parenthesis x right parenthesis. f subscript 2 left parenthesis y right parenthesis space equals 2 over 7 left parenthesis x plus 3 right parenthesis.1 over 7 left parenthesis 1 plus 4 y right parenthesis$ = $2 over 49 left parenthesis x plus 3 right parenthesis left parenthesis 1 plus 4 y right parenthesis$

Since

f(x,y) is not equal to $f subscript 1 left parenthesis x right parenthesis f subscript 2 left parenthesis y right parenthesis$

Hence X and Y are not independent.