# What is the distribution of the ratio of two independent variables, each subject to Rayleigh distribution with different standard deviation?

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.

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

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.