Q: integrand of norm subjected to translation Sorry if my title makes confusion.
Let $\mathbf{x} \in \mathbb{R}^n$ is uniformly distributed on a $(n-1)$-sphere of radius $\sqrt{nP}$, thus $\left\Vert \mathbf{x} \right\Vert^2=nP$.
Obviously,
$$\Lambda=\mathbb{E}_\mathbf{x} \left[ e^{-\frac{1}{2}\left\Vert \mathbf{x} \right\Vert^2} \right]=e^{-\frac{1}{2}nP}$$
With a translation $\mathbf{b} \in \mathbb{R}^n$, the sphere is translated then I wonder if I can calculate the $\Lambda_\mathbf{b}$ ?
$$\Lambda_\mathbf{b}=\mathbb{E}_\mathbf{x} \left[ e^{-\frac{1}{2}\left\Vert \mathbf{x} + \mathbf{b}\right\Vert^2} \right] = \int_\mathbf{x} e^{-\frac{1}{2}\left\Vert \mathbf{x} + \mathbf{b}\right\Vert^2} \mathrm{d}\mathbf{x}$$
Is there any relation between $\Lambda$ and $\Lambda_\mathbf{b}$ ?
