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Generalized length biased distribution is defined as $h\left(x\right)={\phi }_r\left(x\right)f\left(x\right),x>0$, where $f\left(x\right)$ is a probability density function, ${\phi }_r\left(x\right)$ is a polynomial of degree $r$, that is, ${\phi }_r\left(x\right)=a_1(x/{\mu }_1^{\text{'}})+...+a_r(x^r/{\mu }_r^{\text{'}})$, with $a_i>0,i=1,...,r,a_1+...+a_r=1,{\mu }_i^{\text{'}}=E\left(x^i\right)$ for $f\left(x\right),i=1,2...,r$. If $r=1$, we have the simple length biased distribution of Gupta and Keating [1]. First, characterizations of exponential, uniform and beta distributions are given in terms of simple length biased distributions. Next, for the case of generalized distribution, the distribution of the sum of $n$ independent variables is put in the closed form when $f\left(x\right)$...

The bivariate gamma distribution is taken as a life test model to analyse a series system with two dependent components $x$ and $y$. First, the distribution of a function of $x$ and $y$, that is, minimum $(x,y)$, is obtained. Next, the reliability of the component system is evaluated and tabulated for various values of the parameters. Estimates of the parameters are also obtained by using Bayesian approach. Finally, a table of the mean and variance of minimum $(x,y)$ for various values of the parameters involved is...

Samples from the gamma population are considered which are censored both above and below, that is, $r$ observations below and $s$ observations above are missing among $n$ observations. The range in such censored samples is taken up and the distribution of this restricted range is obtained, which can be compared with that in the complete sample case given in a previous paper.