[SOLVED] normal random variable

Jan 2010
18
2
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
Feb 2009
2,763
1,146
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.
 
Jan 2010
18
2
I don't recognize any tricks here, like using the MGF.
what does MGF stand for?

can you please explain this:
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)
 
Last edited:

matheagle

MHF Hall of Honor
Feb 2009
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1,146
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.
 

matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
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?
 
Jan 2010
18
2
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)