# [SOLVED] normal random variable

#### akane

if

$$\displaystyle f_{XY}(x,y) = f_{X}(x)f_{Y}(y) = \frac{1}{2\pi \sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}$$

I have to show that $$\displaystyle Z$$ is also a normal random variable

$$\displaystyle Z = \frac{(X-Y)^2-2Y^2}{\sqrt{X^2+Y^2}}$$

maybe using these substitions

$$\displaystyle A = \sqrt {X^2+Y^2}$$

$$\displaystyle \tan B = \frac{Y}{X}$$

#### matheagle

MHF Hall of Honor
I don't recognize any tricks here, like using the MGF.

One thing for sure is that $$\displaystyle X\sim N(0,\sigma^2)$$ and $$\displaystyle Y\sim N(0,\sigma^2)$$ are independent

So $$\displaystyle \left({X\over \sigma}\right)^2 + \left({Y\over \sigma}\right)^2\sim\chi^2_2$$

IF you're going to transform from X,Y to something you should USE Z and something basic like A equal to X or Y OR maybe your A or maybe A squared? BUT not A and B. Try Z and A.

#### akane

I don't recognize any tricks here, like using the MGF.
what does MGF stand for?

So $$\displaystyle \left({X\over \sigma}\right)^2 + \left({Y\over \sigma}\right)^2\sim\chi^2_2$$
IF you're going to transform from X,Y to something you should USE Z and something basic like A equal to X or Y OR maybe your A or maybe A squared? BUT not A and B. Try Z and A.
this is really hard... I don't have the idea how to do it...

thank you very much (Nod)

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#### matheagle

MHF Hall of Honor
MGF stands for moment generating function.
YOU may have to transform from X,Y to Z,A and then integrate out the A
to obtain the density of Z.

#### akane

MGF stands for moment generating function.
YOU may have to transform from X,Y to Z,A and then integrate out the A
to obtain the density of Z.
What would be A?

#### matheagle

MHF Hall of Honor
What would be A?
You can try any possibility for A.
I was suggesting the one you had already selected, that square root of X and Y squared

Though, I must say, this is a weird problem
I don't see any tricks, are you sure you copied it correctly?

#### akane

You can try any possibility for A.
I was suggesting the one you had already selected, that square root of X and Y squared

Though, I must say, this is a weird problem
I don't see any tricks, are you sure you copied it correctly?
unfortunately I copied it correctly (Crying)
that substitution won't work because I have to rewrite Z as a function of these two new variables, in the case you mentioned I would have Z = Z (X,Y,A)