We are using *Mathematical Statistics with Applications* (7’th Ed.) by Wackerly, Mendenhall and Scheaffer for our calculus based probability and statistics course.

They present the following Theorem (5.5 in this edition)

Let and have a joint density that is positive if and only if and for constants and otherwise. Then $Y_1, Y_2 $ are independent random variables if and only if where are non-negative functions of alone (respectively).

Ok, that is fine as it goes, but then they apply the above theorem to the joint density function: for and 0 otherwise. Do you see the problem? Technically speaking, the theorem doesn’t apply as is NOT positive if and only if is in some closed rectangle.

It isn’t that hard to fix, I don’t think.

Now there is the density function on and zero elsewhere. Here, are not independent.

But how does one KNOW that ?

I played around a bit and came up with the following:

Statement: (note: assume

Proof of the statement: substitute into both sides to obtain Now none of the else function equality would be impossible. The same argument shows that with none of the .

Now substitute into both sides and get but no factor on the right hand side can be zero.

This is hardly profound but I admit that I’ve been negligent in pointing this out to classes.