It is always true that X < Y. Is it possible for X and Y to be independent? Why? X and Y are random variables. It is always true that X < Y. Is it possible for X and Y to be independent? Why?
 A: Yes. For example, $X$ is $0$ or $1$ with equal probability while, independently, $Y$ is $2$ or $3$ with equal probability.
A: Let $\Omega_X$ the set of possible outcomes for $X$ and $\Omega_Y$ for $Y$. Assume both set sare subset of the real numbers.
There are two distinct possibilities:

*

*$\sup \Omega_X \le \inf \Omega_Y$, then $X$ and $Y$ might be independent or not.

*$\sup \Omega_X \ge \inf \Omega_Y$, then $X$ and $Y$ are not independent. Because if  a realization of $X$ is greater than $\inf \Omega_Y$, the outcomes of $Y$ can no longer reach the infimum of the possible outcomes.

You have been given two examples for independence to occur in the first case. The trivial relation $X=Y-2$ for $\Omega_Y=[0,1]$ sets "perfect" dependence between the two and satisifes $X<Y$.
A: A less formal, but more intuitive example:
Consider someone throwing a six sided die, the result being $X$. Then they roll seven six sided dice at once and add the seven rolls. Their sum is $Y$. The first roll is clearly independent of the seven latter rolls, and the seven dice rolls sum to at least $Y=7$, which is greater than the maximum of $X=6$ that can be achieved by the first roll.
