(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given ##f_{X,Y}(x,y)=2e^{-x}e^{-y}\ ;\ 0<x<y\ ;\ y>0##,

The following theorem given in my book (Larsen and Marx) doesn't appear to hold.

2. Relevant equations

Definition

##X## and ##Y## are independent if for every interval ##A## and ##B##, ##P(X\in A \land Y\in B) = P(X\in A)P(Y\in B) ##.

Theorem

##X## and ##Y## are independent iff ##f_{X,Y}(x,y)=g(x)h(y)##.

If so, there is a constant ##k## such that ##f_X(x)=kg(x)## and ##f_Y(y)=(1/k)h(y)##.

3. The attempt at a solution

Consider ##g(x)=2e^{-x}## and ##h(y)=e^{-y}##. Then, ##f_{X,Y}(x,y)=g(x)h(y)##, therefore theorem indicates that ##X## and ##Y## are independent.

The constant ##k## is

##k=\int_0^\infty h(y)dy=1##

##k=\int_0^y g(x)dx = 2(1-e^{-y})##

There is a contradiction in the value of k and it is not constant.

Am I missing something, or is the theorem incomplete in that it is lacking details on the intervals that the random variables are defined on?

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# Homework Help: Independence of Random Variables

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