sum of independent Rayleigh random variables

Question

How do I find the probability density distribution (pdf) of the sum of independent Rayleigh random variables (whose probability density functions are known)? where is the reference? Could anybody please possibly paste the original convolution integral (definition) which expresses the pdf of the sum of independent Rayleigh random variables with different scale parameters? Thanks in advance.

Answer 1

I don't think there's a name for it. For example, according to Maple the sum of two independent Rayleigh random variables with the same scale parameter $b$ has PDF $$ f(z) = \dfrac{z}{2b^2} e^{-z^2/(2b^2)} + \dfrac{\sqrt{\pi} z^2}{4b^3} \text{erf}\left(\frac{z}{2b}\right) e^{-z^2/(4b^2)} - \dfrac{\sqrt{\pi}}{2b} \text{erf}\left(\frac{z}{2b}\right) e^{-z^2/(4b^2)}$$

EDIT: if the scale parameters are $b_1$ and $b_2$, Maple gets $$\eqalign{\sqrt {\pi/2}{{\rm e}^{-1/2\,{\frac {{z}^{2}}{{b_{{1}}}^{2}+{b_{{2}}} ^{2}}}}} {{\rm erf}\left(1/2\,{\frac {\sqrt {2}b_{{2}}z}{b_{{1}}\sqrt {{b_{{1}}}^{2}+{b_{{2}}}^{2}}}}\right)} \left( {z}^{2}-{b_{{1}}}^{2}-{b_{{2}}}^{2} \right) b_{{2}}b_{{1}} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) ^{-5/2}\cr+ \sqrt {\pi/2}{{\rm e}^{-1/2\,{\frac {{z}^{2}}{{b_{{1}}}^{2}+{b_{{2}}}^{2}} }}} {{\rm erf}\left(1/2\,{\frac {z\sqrt {2}b_{{1}}}{b_{{2}}\sqrt {{b_{{1}}}^{2}+{b_{{2}}}^{2}}}}\right)} \left( {z}^{2}-{b_{{1}}}^{2}-{b_{{2}}}^{2} \right) b_{{2}}b_{{1}} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) ^{-5/2}\cr+z{b_{{ 1}}}^{2}{{\rm e}^{-1/2\,{\frac {{z}^{2}}{{b_{{1}}}^{2}}}}} \left( {b_{ {1}}}^{2}+{b_{{2}}}^{2} \right) ^{-2}+{{\rm e}^{-1/2\,{\frac {{z}^{2}} {{b_{{2}}}^{2}}}}}z{b_{{2}}}^{2} \left( {b_{{1}}}^{2}+{b_{{2}}}^{2} \right) ^{-2}\cr} $$