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Mathematics
Set Theory, Logic, Probability, Statistics
Proving a multivariate normal distribution by the moment generating function
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[QUOTE="Torgny, post: 5869170"] [ATTACH=full]213796[/ATTACH] I have proved (8.1). However I am trying to prove that ##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function: ##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## I am trying to use the moment generating function because there is only one moment generating function for a given probability distribution and this also holds for multivariate distributions. But I fail at obtaining a moment generating function. The mgf to ##E(e^{tX_i})## is simply the mgf to the normal distribution but I can't get a moment generating function to ##E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## which from the answer in the text i guess should be a multivariate normal distribution. Can someone help out? [/QUOTE]
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Proving a multivariate normal distribution by the moment generating function
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