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I am trying to find what is the distribution of the ratio of two independent Rayleigh random variables, each of which has different standard deviation.

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  • $\begingroup$ Hello complexfilter, is this a homework problem from a school assignment? You can find some help here. $\endgroup$ Nov 12, 2021 at 23:11
  • $\begingroup$ No. It is related to a project I am working on. I think this paper tells me the answer: epublications.marquette.edu/cgi/… $\endgroup$ Nov 12, 2021 at 23:36

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Wikipedia indicates that the square of a single rayleigh random variable $R$ with parameter $\sigma$ follows a gamma distribution with shape parameter 1 and scale parameter $\frac{1}{2\sigma^2}$, $R^2\sim \text{Gamma}(1,\frac{1}{2\sigma^2})$. Note that $\sigma$ is not the standard deviation. Then the ratio of the square of two Rayleigh random variables can be expressed as the ratio of two chi-square random variables. Under the hypothesis that the two $\sigma$ parameters are equal, this ratio would follow an $F$-distribution. Under the hypothesis that the two $\sigma$ parameters are different, the ratio would follow a non-central $F$-distribution. Your original question asked for the distribution of the ratio, which would be the distribution of a square rooted random variable that follows an $F$-distribution. Let me know if I have made any mistakes or if you need more clarity.

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