Generalized length biased distributions

summary:Generalized length biased distribution is defined as $h(x)=\phi_r (x)f(x), x>0$, where $f(x)$ is a probability density function, $\phi_r (x)$ is a polynomial of degree $r$, that is, $\phi_r (x)=a_1(x/\mu'_1)+ \ldots + a_r(x^r/\mu'_r)$, with $a_i>0, i=1,\ldots ,r, a_1+\ldots + a_r=1, \mu'_i=E(x^i)$ for $f(x), i=1,2 \ldots, r$. 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 $f(x)$ is exponential. Finally, Bayesian estimates of $a_1, \ldots, a_r$ are obtained for the generalized distribution for general $f(x), x>1$