Let Z = X + Y. X follows the Rayleigh distribution given by \begin{equation} f_X(x;\sigma_1) = \frac{x}{\sigma_1^2}\exp\left(-\frac{x^2}{2\sigma_1^2}\right), \; x \geq 0 \label{RayleighPDF} \end{equation} and Y follows the Loglogistic distribution given by \begin{equation} f_Y(y;\sigma_2,\mu) = \frac{\exp(\rho)}{\sigma_2 y \left[1 + \exp(\rho)\right]^2}, \; y \geq 0 \label{LoglogisticPDF} \end{equation} where \begin{equation} \rho = \frac{\log(y) - \mu}{\sigma_2} \label{def_rho} \end{equation} Assuming X and Y are independent, what is the distribution of Z?
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Notation conflict: you have defined $Z$ to be 2 different and conflicting things. Also, you need to specify whether $X$ and $Y$ are independent, or the nature of the dependence. – wolfies Jul 25 '14 at 06:56
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Sorry for the mixed-up. Yes, X and Y are independent (this makes life a lot easier) and z as defined in the last two equations should be given a different symbol, say \rho. – Shen Chiu Jul 25 '14 at 14:20