I have three independent but non-identically distributed random variables $X_1, X_2, X_3$. I define two new random variables for the comparisons of $X_1$ against $X_2$ and $X_3$, call them $P = (X_1 > X_2)$ and $Q = (X_1 > X_3)$. Are $P$ and $Q$ necessarily independent because the $X_i$s are independent?

  • $\begingroup$ I assume you mean $P$ is an indicator function that is 1 if $X_1>X_2$, and 0 else. $\endgroup$ – Michael Sep 26 '17 at 18:38
  • $\begingroup$ Assuming $P$ and $Q$ are indicator functions, the answer is generally "no," you can answer your own question by considering a particular example for which these things are easy to calculate. For example take $X_i$ iid Bernoulli with $P[X_i=0]=P[X_i=1]=1/2$. Intuitively, if you know that $X_1>X_2$, it is more likely that $X_1$ is also larger than $X_3$. $\endgroup$ – Michael Sep 26 '17 at 18:39

No, $P$ and $Q$ are not necessarily independent. A single example suffices:

Let $X_1$ be $0$ with probability $\frac34$ and $100$ with probability $\frac14$.

Let $X_2$ be uniformly distributed on $(10,20)$ and let $X_2$ be uniformly distributed on $(40,46)$.

Then $P$ and $Q$ are completely correlated: If $P$ is true, so is $Q$.

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  • $\begingroup$ Minor typo, you said $X_2$ twice when you meant $X_3$. $\endgroup$ – Michael Sep 26 '17 at 18:44

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