### On the product and ratio of Laplace and Bessel random variables.

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In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability R = Pr(X2 < X1 ) when X1 and X2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R = Pr(X2 < X1 ) has been worked out for the majority of the well-known distributions including Normal, uniform, exponential, gamma, weibull and pareto. However, there are still many other distributions for which the form of R is...

2000 Mathematics Subject Classification: 33C90, 62E99. The Fisher information matrix for three generalized beta distributions are derived.

Data that are proportions arise most frequently in biomedical research. In this paper, the exact distributions of R = X + Y and W = X/(X+Y) and the corresponding moment properties are derived when X and Y are proportions and arise from the most flexible bivariate beta distribution known to date. The associated estimation procedures are developed. Finally, two medical data sets are used to illustrate possible applications.

Differences of two proportions occur most frequently in biomedical research. However, as far as published work is concerned, only approximations have been used to study the distribution of such differences. In this note, we derive the exact probability distribution of the difference of two proportions for seven flexible beta type distributions. The expressions involve several well known special functions. The use of these results with respect to known approximations is illustrated.

The gamma and Rayleigh distributions are two of the most applied distributions in engineering. Motivated by engineering issues, the exact distribution of the quotient X/Y is derived when X and Y are independent gamma and Rayleigh random variables. Tabulations of the associated percentage points and a computer program for generating them are also given.

The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this note, the exact distributions of $\alpha X+\beta Y$, $\left|XY\right|$ and $|X/Y|$ are derived when $X$ and $Y$ are independent normal and logistic random variables. The normal and logistic distributions have been two of the most popular models for measurement errors in engineering.

Beta distributions are popular models for economic data. In this paper, a new multimodal beta distribution with bathtub shaped failure rate function is introduced. Various structural properties of this distribution are derived, including its cdf, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Finally, an application to consumer price indices...

We derive the exact distributions of R = X + Y, P = X Y and W = X / (X + Y) and the corresponding moment properties when X and Y follow Muliere and Scarsini's bivariate Pareto distribution. The expressions turn out to involve special functions. We also provide extensive tabulations of the percentage points associated with the distributions. These tables -obtained using intensive computer power- will be of use to the practitioners of the bivariate Pareto distribution.

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