Related papers: The variance-gamma product distribution

In general, while obtaining the probability density function of sums and products of shifted random variables, ordinary analytical methods such as Fourier and Mellin transforms tend to provide integrals which cannot be expressed in terms of…

Based on a quantum mechanical approach, we investigate moment- (or M-) indeterminate probability densities by way of the characteristic function and self-adjoint operators. The approach leads to new methods to construct classes of…

Let $X_1,\dots,X_n$ be independent normal random variables with $X_i\sim N(\mu_i,\sigma_i^2)$, and set $Z=\prod_{i=1}^n X_i$. We derive asymptotic approximations for the right tail probability $\mathbb{P}(Z>x)$ as $x\to\infty$. When at…

If two random variables X and A are functionally related via f(X)=A for some strictly monotone continuously differentiable function f:R->R, the distribution of X may easily be computed from the distribution of A.

We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\bar{z}) =…

A framework for quantifying dependence between random vectors is introduced. With the notion of a collapsing function, random vectors are summarized by single random variables, called collapsed random variables in the framework. Using this…

We consider two random variables $X$ and $Y$ following correlated Gamma distributions, characterized by identical scale and shape parameters and a linear correlation coefficient $\rho$. Our focus is on the parameter: \[ D(X,Y) = \frac{|X -…

Consider a random sample $X_1 , X_2 , ..., X_n$ drawn independently and identically distributed from some known sampling distribution $P_X$. Let $X_{(1)} \le X_{(2)} \le ... \le X_{(n)}$ represent the order statistics of the sample. The…

This paper studies the properties of the probability density function $p_{\alpha,\nu, n}(\mathbf{x})$ of the $n$-variate generalized Linnik distribution whose characteristic function $\varphi_{\alpha,\nu,n}(\boldsymbol{t})$ is given by…

An integral over the interval $(0,\pi)$ is given for the cumulative distribution function of a sum of independent gamma random variables with different scale and shape parameters. The cumulative distribution function of a positive definite…

The Laplace transforms of the transition probability density and distribution functions for the Ornstein-Uhlenbeck process contain the product of two parabolic cylinder functions, namely D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y),…

In this paper, we derive a probability density function that generalizes the Burr XII distribution. The cumulative distribution function and the $n^{th}$ moment of the generalized distribution are obtained while the distribution of some…

We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the…

The distribution function of the sum $Z$ of two standard normally distributed random variables $X$ and $Y$ is computed with the concept of copulas to model the dependency between $X$ and $Y$. By using implicit copulas such as the Gauss- or…

This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability…

Many models require integrals of high-dimensional functions: for instance, to obtain marginal likelihoods. Such integrals may be intractable, or too expensive to compute numerically. Instead, we can use the Laplace approximation (LA). The…

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include…

We employ optimal control theory to study the problem of estimating the probability density function from a data set originating from an unknown probability distribution. The original variational problem is reformulated as a multi-stage…

In this paper, we determine the density functions of nonsymmetrised doubly noncentral matrix variate beta type I and II distributions. The nonsymetrised density functions of doubly noncentral and noncentral bimatrix variate generalised beta…

To find moments of various estimators related to Autoregressive models of Statistics, one first needs the cumulants of products of two Normally distributed random variables. The purpose of this article is to derive the corresponding…