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If we let $X_{1},...,X_{n}$ be a random sample from pdf. $f(x;\theta)$. Suppose that $X_{(1)}=\min(X_{1},...,X_{n})$ and $X_{(n)}=\max(X_{1},...,X_{n})$. I would like to know that $X_{(1)}$ and $X_{(n)}$ are independent or not?

kjetil b halvorsen
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sg1999
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1 Answers1

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How can they be independent? They never will be. I assume $X_1, X_2, \dotsc, X_n$ are iid rv's, then $$ \DeclareMathOperator{\P}{\mathbb{P}} \P(X_{(1)} > u \mid X_{(n)}=u ) = 0 $$ and unless $\P(X > u)=0$ that is already dependence (this is very easy to see for the very special vase $n=2$). If $n$ is large, the dependence will typically be negligible, and probably there is (for $n \rightarrow \infty$) asymptotic independence, in some sense. But you didn't ask about that.

kjetil b halvorsen
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  • Oh.. I think if $X_{(1)}=X_{2}$, so $X_{(n)}=X_{i}$ for $i=1,3,...,n$, nor $X_{2}$. Then they are independent. I will try to understand that you write. – sg1999 May 16 '17 at 09:26
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    If you know anything about the maximum it tells you something about the minimum. If the were independent this could not happen by definition of independence. Kjetil is correct as his example illustrates. – Michael R. Chernick May 16 '17 at 13:15