Question about Variance propagation of random variables

Hello All,

I have a question about propagation of the variance of random
variables.

This is the problem,

Let X,Y,Z be independent Gaussian random variables with known first
and second order statistics. Namely,

E[x] = E[y] = E[z] = 0
S(x) = Sx, S(y) = Sy, S(z) = Sz (1Sigma standard variation)

So, if we set V = [x; y ; z], we have, Cov[V] = [SxSx 0 0; 0 SySy 0 ;
0 0 SzSz]

Now, consider the simple function, V_Hat = sqrt(X.*X + Y.*Y)

the question is, what is the variance of V_Hat?
Is the following equation correct: S(V_Hat) = sqrt(SxSx + SySy) ?

MATLAB simulations shows that this equation is true for small
variances (SxSx, SySy).

How can I find an Analytical expression for S(V_Hat)?

Thanks in advance,
Miki


.

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