Related papers: A note on the distribution of the product of zero …
Over the last 80 years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a…
We prove that the distribution of the product of two correlated normal random variables with arbitrary means and arbitrary variances is infinitely divisible. We also obtain exact formulas for the probability density function of the sum of…
Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $\sigma_{X_i}^2$, $i=1,\ldots,M$, and $\sigma_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$.…
The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas.…
We derive the exact probability density function of the product of $N$ independent variance-gamma random variables with zero location parameter. We then apply this formula to derive formulas for the cumulative distribution function and…
Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product $XY$ is derived. Some basic distributional properties are also derived, including…
We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From…
Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including…
We obtain new closed-form formulas for the moments and absolute moments of the variance-gamma distribution. We thus deduce new formulas for the moments and absolute moments of the product of two correlated zero mean normal random variables.
The wrapped normal distribution arises when a the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an…
Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These…
We represent the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, as a difference of two independent noncentral chi-square random variables (which we refer to as…
The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the…
We derive a closed-form expression for the orthogonal polynomials associated with the general lognormal density. The result can be utilized to construct easily computable approximations for probability density function of a product of…
We derive a fully analytical, one-line closed-form expression for the cumulative distribution function (CDF) of the product of two correlated zero-mean normal random variables, avoiding any series representation. This result complements the…
Lognormal random variables appear naturally in many engineering disciplines, including wireless communications, reliability theory, and finance. So, too, does the sum of (correlated) lognormal random variables. Unfortunately, no closed form…
We derive a simple and precise approximation to probability density functions in sampling distributions based on the Fourier cosine series. After clarifying the required conditions, we illustrate the approximation on two examples: the…
Several formulations have long existed in the literature in the form of continuous mixtures of normal variables where a mixing variable operates on the mean or on the variance or on both the mean and the variance of a multivariate normal…
This paper develops a polynomial normal transformation model, whereby various non-normal probability distributions can be simulated by the standard normal distribution. Two methods are presented to determine the coefficients of polynomial…
In this paper we discuss the representation of the joint probability density function of perfectly correlated continuous random variables, i.e., with correlation coefficients $\rho=pm1$, by Dirac's $\delta$-function. We also show how this…