Ratios X/ZX/Z, Y/ZY/Z built from independent random variables (X,Y)(X,Y) and ZZ may not always be dependent

It is a commonly held “belief” in many quarters that the ratios U=XZ,V=YZ are necessarily dependent random variables when the random vector (X,Y)(X,Y) is independent of the random variable ZZ simply because both U,VU,V involve ZZ. Any outpouring support behind such “belief” often gets louder when (X,Y)(X,Y) are assumed dependent. The purpose of this note is to emphasize that such “beliefs” may be false. Concrete examples are given when (i) X,YX,Y are independent but U,VU,V may be dependent or independent, (ii) X,YX,Y are dependent but U,VU,V may be dependent or independent. Finally, a simple general approach is given for beginners without exploiting joint and/or conditional densities.