-1
$\begingroup$

Let $X,Y \sim U_{[-1,1]}$ are independent random variables. Let us define $U=\max (X,Y)$ and $V=\min(X,Y)$. Are random variables $U,V$ independent ?

$\endgroup$
  • $\begingroup$ Intuitively, does knowing the value of $U$ tell you any information about $V$? $\endgroup$ – angryavian Dec 3 '18 at 22:16
  • $\begingroup$ I think, that yes. $\endgroup$ – PabloZ392 Dec 3 '18 at 22:19
  • $\begingroup$ I think, that $U,V$ are dependent, but I have problem, because I trying find an example. $\endgroup$ – PabloZ392 Dec 3 '18 at 22:27
3
$\begingroup$

$P(V > 0) = P(X > 0 \wedge Y > 0) = 0.5 \times 0.5 = 0.25$

However, $P(V > 0 | U = 0) = 0$, since $U = 0 \implies X \leq 0 \wedge Y \leq 0$.

i.e. the distribution of $V | U$ is not the same as the unconditional distribution of $V$, so they are not independent.

$\endgroup$
  • $\begingroup$ $P(V > 0) = P(X > 0 \color{red}{\wedge} Y > 0) = 0.5 \times 0.5 = 0.25$. Is this correct? $\endgroup$ – PabloZ392 Dec 3 '18 at 22:41
  • $\begingroup$ Yes - the event $min(X, Y) > 0$ is the same as the event $X, Y > 0$, i.e. the minimum is greater than zero if both of the values are greater. $\endgroup$ – ConMan Dec 3 '18 at 22:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.