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.