I am attempting to prove that the multivariate distribution with maximum entropy for a given covariance is a Gaussian. (PRML, Bishop, problem 2.14). Bishop suggests the use of Lagrange multipliers - concretely, that should maximize $$ \text{H}[x] = -\int p(x)\log(x)dx $$ subject to the constrain...

Description of context Given are independent random variables $n,u$ that are normally and uniformly distributed, $$n\sim\mathcal{N}_{\mu,\sigma}=\frac{1}{\sigma\sqrt{2\pi}}\text{exp}\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)=f(x)\ \ \ \ \ \text{ for } x\in\mathbb{R}\\ u\sim \ma...

I have a positive integer $n$, and a multiset $S$ of positive integers. $S$ has $n$ elements. For all $s \in S$, $s$ is a divisor of $n$. I believe that there must exist a subset (submultiset) $S' \subset S$ such that the elements of $S'$ sum to $n$. For example, $n$ could be 6, and $S$ could be ...

I have a very simple question but it's confusing me a bit nonetheless. Suppose $(\Omega,\mathcal F,\mu)$ be a measure space and $f\geq0$ is a measurable function. Let $A=\{\omega\in \Omega:f(\omega)>0\}$ and suppose $\mu(A^c)=0$. Suppose $g:\Omega\to\mathbb R_+$ is defined by $g(\omega)=1/f(\o...

There is a big discussion in our class. Teacher says, $f:\mathbb Z^{+}\rightarrow \left\{0\right\}$ and $f(x)=0$ is a constant function. But, student says only $f:\mathbb R\rightarrow \left\{0\right\}$ and $f(x)=0$ is a constant function or $$f(x)=0\,\,\forall x\in\mathbb R$$ Unfortunately, wik...