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Given a random variable $X$, we denote by $X_1, X_2, \dots$ a sequence of iid copies of $X$. Question: Does there exist a random variable $X$ with $\mathbb E[X^+] = \mathbb E[X^-] = +\infty$, but $$\l …

Let $X, Y$ be non negative random variables with finite expectation. We say that $Y$ stochastically bounds $X$ if there exists some $C > 0$ such that for all $x \in \mathbb R$, $$\mathbb P(X \geq x) \ …

Let $a, n$ be positive integers with $a < n$. A revolver with $n$ chambers is loaded with $a$ bullets, where the distribution is uniform among all $\binom{n}{a}$ possible choices of $a$ objects fillin …

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Define, for each $n$, the order statistic $O_n$ of $X_n$ by $$O_n := \frac{1}{n}\#\{1 \leq k \leq n \, \, | \, X_k \ …

Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$. Question: Let $N \geq 2$ be an arbitrary in …

We recall that a coupling of probability distributions $\mu_1, \dots, \mu_n$ on $\mathbb R$ is a set of random variables $X_1, \dots, X_n$ defined on the same probability space such that $X_i$ is dist …

The Pareto Principle roughly states that in many societies, the top 20% of people hold over 80% of the wealth. Suppose we had a society that satisfied this principle in every stratum of society - how …

Recall that a coupling of probability measures $\mu_i$ is a set of random variables $X_i$ defined on the same probability space $\Omega$ such that $X_i \sim \mu_i$. Question: Let $\mu_1, \dots, \mu_n$ …

Just a fun problem I thought of. A man is playing a magical pipe organ - every chord is an integer number of decibals (dB) loud. The softest chord is $0$ dB. Every chord of $N > 0$ dB creates a rando …

For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$. Question: Is it true that $\mathbb E[|\text{exp}\, (\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $T …

Note: Here all processes take values in $[0, 1]$. Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant. Let $X$ be the solution to the SDE $$dX_t = \sigma X_t \, dW_t$$ w …

Let $X$ be a geometric Brownian motion, satisfying the SDE $$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$ for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant. Define the drawdown …