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Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest: $1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed; $...

Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...

I am interested in maximizing the variance of a random variable $X$ supported on $[0,1]$. Formally, $$\max_{P_X: X \in [0,1]} {\rm Var}(X),$$ where $P_X$ is a distribution of $X$. This question is ...

The following problem is common in financial economics $$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$ That is, given a random variable $y(\theta)$ ($\...