- #1
showzen
- 34
- 0
Homework Statement
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.
Homework 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)##.
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?