Mark ( [EMAIL PROTECTED]) write:
> I have a problem that puzzles me. It's a theorem that is listed in an
> inference book. Here it is:
>
> If a random sample with size two is taken from a distribution with
> positive variance and if the sum and the difference of the two
> components of that sample are independent, then the distribution from
> which the sample is taken is a normal distribution.
>
> Could anybody tell me how to proceed in order to prove that?
>
Feller (1971, An Introduction to Probability Theory and Its Applications,
Vol II, 2 Ed.) proves a more general result in Section III.4 (pp. 77-80).
Let Y1 = a X1 + b X2 and Y2 = c X1 + d X2. Feller shows:
Suppose that X1 and X2 are independent of each other, and that the same is
true of the pair Y1, Y2. If no coefficient a,b,c,d vanishes then all four
variables are normal.
gary
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