I am trying to find $Pr\{X > \alpha\}$ when $X$ is a nonnegative combination of independent Rayleigh random variables. Ie:

$$ X = \displaystyle\sum\limits_{i=1}^{N} a_{1} X_{1} $$ $$ X_{i} \sim R(\sigma_{i}), \quad a_{i} \in \mathbb{R_{+}} \; \forall i$$

Starting Case $N=1$:

$$X_{N_{1}} = a_{1} X_{1}$$

Thus the characteristic function of $X$ is:

$$ \psi_{X_{N_{1}}}(t) = \psi_{X_{1}}(a_{1}t)$$ $$ \psi_{X_{N_{1}}}(t) = 1-\sigma_{1}a_{1}t e^{-\frac{\sigma_{1}^{2}a_{1}^{2}t^{2}}{2}} \sqrt{\frac{\pi}{2}}\bigg(erfi\Big(\frac{\sigma_{1}a_{1}t}{\sqrt{2}}\Big) -i\bigg)$$

Thus, $X_{N_{1}} \sim R(a_{1}\sigma_{1})$.

Second Case $N=2$:

$$X_{N_{2}} = a_{1}X_{1} + a_{2}X_{2}$$


$$\psi_{X_{N_{2}}}(t) = \psi_{X_{1}}(a_{1}t) \psi_{X_{2}}(a_{2}t)$$ $$\psi_{X_{N_{2}}}(t) = \Bigg(1-\sigma_{1}a_{1}t e^{-\frac{\sigma_{1}^{2}a_{1}^{2}t^{2}}{2}} \sqrt{\frac{\pi}{2}}\bigg(erfi\Big(\frac{\sigma_{1}a_{1}t}{\sqrt{2}}\Big) -i\bigg)\Bigg) \cdot \Bigg(1-\sigma_{2}a_{2}t e^{-\frac{\sigma_{2}^{2}a_{2}^{2}t^{2}}{2}} \sqrt{\frac{\pi}{2}}\bigg(erfi\Big(\frac{\sigma_{2}a_{2}t}{\sqrt{2}}\Big) -i\bigg)\Bigg) $$

This doesn't seem to pan out to a Rayleigh type characteristic function. So, I have two questions:

  1. How is $X$ distributed?
  2. How can I find $Pr\{X>\alpha\}$? How could I do it numerically if there is no closed form answer to how $X$ is distributed?

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