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The non-central chi-square distribution with $\nu$ degrees of freedom has a density which can be expressed as $$ f(x) = \sum_{i=0}^{\infty} c_i f_{\nu + 2i}(x), $$ where $c_i$ is a function of the non …

Let $\mathcal{X}$ be some set of independent random variables. Define the ordering on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there …

Let $\vec{x}$ and $\vec{y}$ be zero mean $n$-variate Gaussian variables with covariances $\Sigma^x, \Sigma^y$. Suppose they have identical marginals ($\sigma_{i,i}^x = \sigma_{i,i}^y$ for all $i$), an …

Suppose vector $R$ is a random permutation of the integers 1 through $n$ such that $$ \mathcal{P}\left(R_i = 1\right) = \pi_i, $$ for given vector of probabilities $\pi$. Moreover, assume a 'proport …

Let $x$ be a continuous random variable with zero mean and zero skew. What are the conditions under which we can say that $x$ can be expressed as the product $z y$ where $z$ is a standard normal and $ …