7
votes
1answer
289 views

Probability density that minimizes the sample range

Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in ...
1
vote
1answer
71 views

Numerical optimisation for multivariate Gaussians

Hi, I want to calculate $ f_{\mathbf x}(x_1,\ldots,x_k)\, = \frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf ...
0
votes
1answer
238 views

Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]

I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
7
votes
2answers
309 views

Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...
8
votes
2answers
337 views

An extension of Gaussian Isoperimetry

The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...