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I want to know the proof of the following statement about normal distribution:

If the sample mean and sample variance are independent for a population, then the distribution of the population is normal.

I cannot find the proof in any probability/statistics textbook that I have. Please help.

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    @Michael Hardy: Yes, I think independence is true for all sample sizes in order to get the assertion. I forgot where I see this assertion.2012-02-20

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See this question on Stats.SE

There are articles quoted here but no proofs shown. Wikipedia also quotes an article.

So far I haven't found the proof

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    I see. Not sure if any textbook contains the proof, besides the two papers referenced therein.2012-02-20