Is there a name for the distribution $p(x) = \exp( x \cdot u ) / Z$?

Question

Is there a name for the distribution $$ p(x) = \exp( x \cdot u ) / Z $$ where $Z$ is the normalizing constant and $u = E[X]$? $X$ is a standard normal random variable.

Answer 1

Yes, there is!

Let $X$ be a random vector in $\mathbb{R}^n$ with probability density $f$ and mean $\mu = \mathbb{E}[X]$. Assuming that its moment generating function $M_X(u):=\mathbb{E}\left[e^{u^\top X}\right]$ is such that $M_X(u)<\infty$, one can define the exponential tilting of $X$ as a random variable $X_u$ with probability density

$$f_{u}(x) = \frac{e^{u^\top x}f(x)}{M_X(u)}.$$

In your case, you are thus looking at the exponential tilting of $X$ with $u=\mu$. Note also that the normalization constant $Z$ in your description is nothing but the moment generating function of $X$.