# When is a function of two Normal variables Normal?

I know that for two independent Gaussian variables, the sum and the product is Gaussian as well.

Is there a general form for this, ie a class of functions $f:\mathbb R^2 \rightarrow \mathbb R$ so that $f(X,Y)$ is Gaussian when X,Y are independent Gaussian (or possibly - jointly Normal)?

Thanks.

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... Who told you that the product of two independent Gaussian variables is Gaussian? – user1551 Jan 3 '13 at 9:40
I saw this: tina-vision.net/docs/memos/2003-003.pdf but maybe I was wrong... – yoki Jan 3 '13 at 22:33
I see. What it means is actually that the product of two Gaussian p.d.fs is Gaussian. Not that the product of two Gaussian distributions is Gaussian. I must say the wording of the authors of your cited document is inappropriate. – user1551 Jan 4 '13 at 9:26

The product of two (independent) Gaussian random variables is not necessarily a Gaussian random variable! (See here).

Here are some results:

Let $X:=(X_1,\ldots,X_n) \sim N(\mu,C)$ jointly normal.

1. Let $b \in \mathbb{R}^m$, $A \in \mathbb{R}^{m \times n}$. Then $$Y:=(Y_1,\ldots,Y_m) := A \cdot X+b$$ is normal, precisely $$Y \sim N(b+A \cdot \mu, A \cdot C \cdot A^T)$$
2. (Special case of 1.) $$Y:=\ell^T \cdot X = \sum_{i=1}^n \ell_i \cdot X_i$$ is normal where $\ell \in \mathbb{R}^n$ arbritary, precisely $$Y \sim N(\ell^T \cdot \mu, \ell^T \cdot C \cdot \ell)$$ In particular ($n=2$): $a \cdot X_1 + b \cdot X_2$ is normal for all $a,b \in \mathbb{R}$.

Now let $X,Y$ normal and independent random variables. Then $(X,Y)$ is a normal random variable (so you can apply 1. and 2. to $(X_1,X_2):=(X,Y)$).

Since you asked in particular for functions $f: \mathbb{R}^2 \to \mathbb{R}$: Let $$f(x,y) := a \cdot x + b \cdot y + c \qquad (a,b,c \in \mathbb{R})$$ then $f(X,Y)$ is Gaussian.

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Hmm, why post two results when (1) the same as (2) with $A\to\ell^T$, $b\to0$? Correspondingly, the variance of $Y$ there should be $\ell^TC\ell$ and not $\ell^TC\mu$. – Rahul Jan 3 '13 at 10:44
@RahulNarain Sorry, there was a typo in it. And yes, you are right (1) is a special case of (2), but since it's an important one, I thought it would be a good idea to mention it explicitely. I rewrote it... – saz Jan 3 '13 at 10:51
Hi. The question is more when these two variables are NOT multivariate Gaussian, but when they are just some normal variables. – yoki Jan 7 '13 at 11:22
@ido You wrote "independent Gaussian" above, that's why I thought you were looking for something like this. If $X$, $Y$ are normal (but not jointly normal, in particular not independent), I would say there's no such general class of functions. Of course, if you would fix $X$ and $Y$, then you could probably find a function $f$ such that $f(X,Y)$ is normal - but this function $f$ depends on the properties of $X$, $Y$ resp. $(X,Y)$ – saz Jan 7 '13 at 16:21
You're right, sorry, I got confused with another question... – yoki Jan 7 '13 at 20:40