Power of Multivariate Gaussian density Let $X$ be a random variable following a multi-variate Gaussian distribution: $X \sim N\left(\mu,\Sigma^2\right)$ where $\mu$ and $\Sigma^2$ are the mean vector and covariance matrix. Let $f(x)$ be the density function of $N\left(\mu,\Sigma^2\right)$ and $\alpha$ a positive real number. If $f(x)$ is raised to the power of $\alpha$, is $(f(x))^\alpha$ still a density function for some multi-variate Gaussian distribution? If yes, and assume the pdf of $Y$ is $(f(x))^\alpha$. What are the mean and covariance of $Y$ in terms of $\mu$ and $\Sigma^2$?

Taking the density $f$ to the power $\alpha$ gives another Normal density with covariance matrix $\alpha^{-1}\Sigma$,
$$f^\alpha(x;\mu,\Sigma)=(2\pi|\Sigma|)^{-\alpha/2}\exp\left\{
-\alpha(x-\mu)^\mathsf{T}\Sigma^{-1}(x-\mu)/2\right\}$$
except that the normalising constant is wrong:
$$(2\pi|\Sigma|)^{\alpha/2} \ne (2\pi\alpha^{-1}|\Sigma|)^{1/2}$$
