Let ${\bf{x}}\sim {\cal{N}}({\bf{\mu}},\sigma^2{\bf{I}}) \in \mathbb{R}^n$
I am wondering if I can find an analytical expression for the expectation and covariance of ${\bf{x}}$ given that $\frac{||\bf{x}||^2}{\sigma^2}$ crosses some threshold $\gamma$:
- $\mathrm{E}[{\bf{x}}| \frac{||\bf{x}||^2}{\sigma^2}> \gamma ]$
- $\mathrm{Cov}[{\bf{x}}| \frac{||\bf{x}||^2}{\sigma^2}> \gamma ]$.
I notice that $\frac{||\bf{x}||^2}{\sigma^2}\sim \chi^2_n(\lambda)$ where the non-centrality parameter is $\lambda=\frac{\bf{\mu^T\mu}}{\sigma^2}$.
So far, I managed to show that for the case where $\bf{\mu}=0$ $\mathrm{E}[{\bf{x}}| \frac{||\bf{x}||^2}{\sigma^2}> \gamma ]=0$ and that $\forall i\neq j: \mathrm{E}[x_ix_j | \frac{||\bf{x}||^2}{\sigma^2}> \gamma ]=0$.
Any help would be appreciated.