(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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Independence of Random Variables

Have something to add?

**Physics Forums | Science Articles, Homework Help, Discussion**