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How can I show that if $X_1$ and $X_2$ are independent random variable having a common distribution function $F$, then linear combinations of $X_1$ and $X_2$ are pairwise independent if and only if $F$ is a normal distribution function?

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  • $\begingroup$ I would not expect $4X_1+5X_2$ to be independent of $5X_1+4X_2$ whether or not $X_1$ and $X_2$ are normally distributed. Perhaps I have misunderstood the question $\endgroup$ – Henry Nov 28 '17 at 13:16
  • $\begingroup$ You are right. I think about the case where for example we have two not equal linear combinations. $\endgroup$ – Amolika Nov 29 '17 at 12:58

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