If XYZ are rvs such that X and Y are independent and Y and Z

If X,Y,Z are r.v.s such that X and Y are independent and Y and Z are independent, does it follow that X and Z are independent?

Hint: Think about simple and extreme examples.

Solution

Simply take Z = 2X, if X and Y are independent, clearly 2X and Y are also independent => Z and Y are independent

But X and Z are not independent!

For example, take X = number appearing on tossing a fair die. Let Y = number appearing on tossing another fair die. X and Y are independent. Y,Z are independent because Z=2X, but X and Z are not independent because Z=2X