Proving a multivariate normal distribution by the moment generating function

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Torgny 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 cant 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?

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FactChecker
Gold Member
Torgny
Thanks! But what do I do with:

##-\frac{1}{n}##
in

##E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})##

I can see that the rest follows the relation:

##\varphi_{X+Y}=\varphi_{X}\varphi_{Y}##

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StoneTemplePython
Gold Member
I'm not quite sure of your motivation for what you're trying to do here. Moment generating functions can be useful, but frequently are not needed -- this is one of those cases.

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It seems to me there are only two building blocks.

1.) "since the sum of independent normal random r.v.'s is a normal r.v." This holds -- the sum of finitely many independent normal rvs is normal rv. It is enough to prove that the convolution of two independent normals is a normal, and induct from there. Moment generating functions aren't needed here.

2.) the fact that the joint distribution of normal r.v.'s is a multivariate normal r.v.

If you have proven 1, apply it such that ##Y_i := X_i - \overline{X}## must be a normal r.v. Why? Because it is the convolution of ##X_i## with 1 piece (with ##\frac{1}{n}## weighting) that is strictly dependent on ##X_i## -- and in fact is a negative scaled down, version of ##X_i##. This has the effect of rescaling ##X_i##, but its still a normal r.v.. Then the resulting##X_i## is convolved with ##(n-1)## other independent normals (though each value of said n-1 normals is rescaled by -1). So repeatedly apply part 1 here, and the result is a normal r.v.

Then apply 2.

Torgny
I'm not quite sure of your motivation for what you're trying to do here. Moment generating functions can be useful, but frequently are not needed -- this is one of those cases.

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It seems to me there are only two building blocks.

1.) "since the sum of independent normal random r.v.'s is a normal r.v." This holds -- the sum of finitely many independent normal rvs is normal rv. It is enough to prove that the convolution of two independent normals is a normal, and induct from there. Moment generating functions aren't needed here.

2.) the fact that the joint distribution of normal r.v.'s is a multivariate normal r.v.

If you have proven 1, apply it such that ##Y_i := X_i - \overline{X}## must be a normal r.v. Why? Because it is the convolution of ##X_i## with 1 piece (with ##\frac{1}{n}## weighting) that is strictly dependent on ##X_i## -- and in fact is a negative scaled down, version of ##X_i##. This has the effect of rescaling ##X_i##, but its still a normal r.v.. Then the resulting##X_i## is convolved with ##(n-1)## other independent normals (though each value of said n-1 normals is rescaled by -1). So repeatedly apply part 1 here, and the result is a normal r.v.

Then apply 2.

Thanks for the insight. However I believe that I will not get an accepted answer unless I prove it mathematically. For 1) I can prove it as noted above like this:

##E(e^{X+Y})=E(e^X)E(e^Y)##

But I don't know how to prove the things you adress afterwards with equations.

StoneTemplePython
Gold Member
Thanks for the insight. However I believe that I will not get an accepted answer unless I prove it mathematically. For 1) I can prove it as noted above like this:

##E(e^{X+Y})=E(e^X)E(e^Y)##

But I don't know how to prove the things you adress afterwards with equations.

They also show the convolution of two normal r.v.'s directly here:

https://ocw.mit.edu/courses/electri...s-convolution-correlation/MIT6_041F10_L11.pdf

(MIT is not very big on moment generating functions. )
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To be clear the outline I gave was mathematical. You'd just need to recut it into a couple of lemmas, then carefully use induction in the main argument. The underlying idea that comes up over and over (in both part 1 and part 2) is that convolving a random variable with a scaled down version of itself is just a rescaling. And convolving a normal r.v. with an independent normal r.v. results in a normal r.v.

There would only be one or two equations here -- and it has a linear algebra flair in that everything we're interested in is written as a linear combination of a scaled version of identical random normals (i.e. rescaling) and independent normals. It's actually a very simple idea.

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You seem to be quite keen on using MGFs which is not how I'd look at this. Good luck.