Suppose that $X_1,...,X_n$ are Random Variables and given that there exist an $k$ where k is an integer and $1\le k\le n-1$ s.t. the joint distribution $F_{X_1,...,X_k}$ are independent to $F_{X_k+1,...,X_n}$, prove that for all $1\le r \le k\le m\le n-1$ the joint distribution of $X_1,...,X_r$ is independent to joint distibutions $X_{m+1},...,X_n$

  • $\begingroup$ What did you try? $\endgroup$ – Did Oct 4 '12 at 13:54
  • $\begingroup$ i tried to use contradiction, but not sure how to get the contrary $\endgroup$ – Mathematics Oct 4 '12 at 14:04
  • $\begingroup$ I fail to see how a proof by contradiction would help. More to the point: what is the conclusion you try to reach, that is, what are you trying to prove? $\endgroup$ – Did Oct 4 '12 at 16:59

You could try to use the fact that if $X$ and $Y$ are independent random variables, and if $f$ and $g$ are two (measurable ...) functions, then also $f(X)$ is independent from $g(Y)$.

  • $\begingroup$ but here is about the the joint pmf not pmf of a random variable $\endgroup$ – Mathematics Oct 8 '12 at 8:36
  • $\begingroup$ kjetil: +1. $ $ $\endgroup$ – Did Oct 8 '12 at 10:03

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