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Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda_1,...,\lambda_n$ be given real numbers. What is the distribution of $$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$ Does it happen …

Suppose a random positive definite matrix $A\in\mathbb{R}^{n\times n}$ has density function (with respect to the lebesgue measure on $\mathbb{R}^{n(n+1)/2}$) $f(A)=g(\lambda_1(A),...,\lambda_n(A))$ w …

Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, …

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 $l$- …

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 …

Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that th …

Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?

$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, …

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 …