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Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$? $$\mathbb{P}(X\geq x)$$ what is the order of the above probability as $x\to+\infty$?

What is the order of the following expectation with respect to $n$?: $$\mathbb{E}(\min_{1\leq i\leq n}|z_i|^2)$$ where $$(z_1,...,z_n)^T\sim N(0,I+11^T), 1=(1,1,...,1)^T$$ I know that when $z_i$ are …

Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order? $$\mathbb{P} …

Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$- …

Is there a simple upper bound of the following fraction of Gamma functions for any $a,b\geq1/2$: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$ An upper bound in the following form is …

Can we solve the following recurrence relation: $$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$ with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$ I encountered this …

We consider the asymptotics of the empirical distribution of eigenvalues of $S_{n,p}$. We let $p/n\to c\in(0,1)$ where $c$ is a constant. …

Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean. Let $X\sim\exp(\lambda)$ where the de …