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Depends on the relationship between $ x $, $ k $, and $ n $ and the distribution. For example consider whether $ x $ is close to $ k/n $, if the distribution is uniform.

Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < p,p' < 1$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful upper bounds would be helpful. …

You can put the uniform distribution on any nonempty finite set whatsoever. In this case, you could call it the uniform distribution on the set of nonnegative integer solutions to an equation.

$$ f(t,s)= f(t\mid s)\cdot f(s) = \frac1{\theta^k\Gamma(k)} s e^{-st}\cdot s^{k-1}\cdot e^{-s/\theta} $$ $$ = \frac1{\theta^k\Gamma(k)} s^{k}\cdot e^{-st-s/\theta} $$ for $s>0$, $t>0$. Here $s=\lambda …

In other words, given a matrix $(p_{ij})$, is there a distribution on partitions of $[n]$ with the above property, and if so is there a way to sample from such distribution? No; $ p_{1,2}=p_{1,3} …

Here's a positive answer for the binomial distribution in the case $p=1-\frac1{n}$. (I suppose you could check some other cases like $p=1-\frac2n$ in a similar way.) We need to prove $$\binom{n}{n} …

For any $n$, $$\Pr(Y\le n)=\Pr(X_i\le n\,(\forall i))=\prod_i\Pr(X_i\le n)=\lim_{i\to\infty}\Pr(X_1\le n)^i=0.$$ So $Y=\infty$ with probability 1.

How about if we let $f(t)=\int_0^t W_s^2ds$ where $W$ is a standard Wiener process, and $g(t)=\int_0^tf(s)ds$. Then $g''(t)=W_t^2\ge 0$ so $g$ is convex. For general $C$ replace 0 by $\inf C$.

It just means that $$\mu\left(\{f: K_f>y\}\right)<\frac{\alpha}{y^\beta}$$ and $$\mu\left(\{f: \rho(x_0,f(x_0))>y\}\right)<\frac{\alpha}{y^\beta}$$

This answer is referring to version 1 of the question. What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniform …

You can use maximum likelihood estimation: https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation

Let $A_t$ be the event that $S_1$ is a substring of $S_2$, $S_2=pS_1q$, where the length of $p$ is $t$. Then the probability of $\cup_t A_t$ can be found by inclusion-exclusion as $$\sum P(A_t)-\sum P …

It's not possible. Let $X$ be constant equal to 1. Let $B_1,B_2,B_3$ be independent Bernoullis. Let $R_1=B_1+B_2$, $C_1=B_3$. Let $R_2=B_1$, $C_2=B_2+B_3$. Then $R_1X+C_1=R_2X+C_2$. So even if yo …

Here are some interesting theoretical notions of distance between two such distributions: Wasserstein distance Lévy–Prokhorov metric Total variation distance of probability measures

It seems that in general this is an almost arbitrarily hard problem. Consider the simpler countably infinite case $X=\mathbb N$, $Y=\{0,1\}$. Thus $H=\{x:h(x)=1\}$ is a "random set". Fix $g:X\to\{0, …