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I have a gaussian random variable $\xi \sim \mathcal{N} (\mu,\sigma^2)$, and the following function $g(\xi)$: \begin{equation} g(\xi)=-( \xi +\textrm{b}^{\textrm{T}} x) + \left\lVert{\begin{matrix} \xi +\textrm{c}^{\textrm{T}} x \\ \textrm{d}^{\textrm{T}}x \end{matrix}}\right\rVert \end{equation}

I need to obtain the mean and standard deviation of $g(\xi)$. I know that I can calculate them using the definitions: \begin{equation} \operatorname{E} [g(\xi)]=\int_{-\infty}^{\infty} g(\xi) f(\xi) \,d\xi \end{equation} \begin{equation} \operatorname{Var}[g(\xi)]=\textrm{E} [(g(\xi)-\textrm{E}[g(\xi)])^2] \end{equation}

But this method is quite cumbersome. Is there any other method I can use?

Michael Hardy
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luisba
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  • $ \textrm{Var}[g(\xi)]=\textrm{E} [\left(g(\xi)-\mathrm{E}(g(\xi)\right))^2] $ – user158565 Nov 02 '18 at 18:22
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    For the variance have you considered the Delta Method: https://en.wikipedia.org/wiki/Delta_method ? – JimB Nov 02 '18 at 18:42
  • A [saddlepoint approximation](https://stats.stackexchange.com/questions/191492/how-does-saddlepoint-approximation-work/191781) ought to work well. – whuber Nov 02 '18 at 19:24
  • To make sure I understand your notation, is $g(\xi)$ equivalent to $-(\xi+b)+\sqrt{(c+\xi )^2+d^2}$ where $b$, $c$, and $d$ are known constants? – JimB Nov 03 '18 at 22:41
  • I agree that my formulation was confusing. This formulation comes from an optimization problem where $x$ are decision variables, but from the random variable perspective they can be regarded as constants, just like in your formulation. Note that the square root in your formulation must be positive, since the norm enforces nonnegativity – luisba Nov 04 '18 at 11:57
  • One last clarification: while a symbol approximation would certainly be desired, is it essential to your needs? – JimB Nov 05 '18 at 18:53

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