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I need to understand the Rayleigh distribution for a homework assignment in computer networks. Unfortunately, I lack the background knowledge in the field of statistics and probability theory to understand the descriptions that are given elsewhere about the Rayleigh Distribution, as it leads me from one page to another and I get more confused.

If you could use an example to illustrate the case it would be even more useful.

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    $\begingroup$ If you could say a little about what you understand thus far, & what parts you're confused about, that would help. $\endgroup$
    – gung - Reinstate Monica
    Commented Jul 12, 2014 at 14:04
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    $\begingroup$ Your question is broad and unspecific. What do you need to understand about it? $\endgroup$
    – Glen_b
    Commented Jul 12, 2014 at 14:32
  • $\begingroup$ in the context that I am asking it is related to the Rayleigh Fading from the field of communications systems. I would like to know what is the idea behind it? What phenomena does it tell? I think it was related to random independent variables etc. $\endgroup$
    – Kristof Tak
    Commented Jul 12, 2014 at 16:47

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In the context of communication systems, Rayleigh random variables arise as the amplitude of received signals. A model for such a signal is $$X\cos(\omega_0 t)- Y \sin(\omega_0 t) = R \cos(\omega_0 t + \Theta)\tag{1}$$ where $X$ and $Y$ are independent Gaussian random variables with the same variance $\sigma^2$, which can be expressed as the right side of $(1)$ with $R$ being a Rayleigh random variable and $\Theta$ uniformly distributed on $[0,2\pi)$; $R$ and $\Theta$ are independent too. The intuitive explanation for the model is that a transmitted signal $A\cos(\omega_0 t)$ is reflected off many scatterers resulting in a received signal that is formed by the sum of many tiny (small-amplitude) reflections. The Central Limit Theorem then allows us to pretend that the resulting sum as a Gaussian random variable. The right side of $(1)$ should also be familiar to statisticians as the linchpin of the Box-Muller method for generating samples of Gaussian random variables.

If your work in computer networks deals with reliability of systems and networks, then you should know that if a hazard rate is assumed to be increasing linearly with time, then the lifetime of the system is a Rayleigh random variable.

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    $\begingroup$ I think your second example, with the hazard rate, is terrific and something I did not know $\endgroup$
    – Cam.Davidson.Pilon
    Commented Jul 13, 2014 at 12:44
  • $\begingroup$ @DilipSarwate Amazing explanation! Thanks a lot. Could you provide me some pointers which explain in the same level of quality yet simply the Central Limit Theorem and what a Gaussian Random Variable means? I wonder if there is a chance I can convince to extend your answer and include them in the explanation :) $\endgroup$
    – Kristof Tak
    Commented Jul 13, 2014 at 16:58
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    $\begingroup$ Wolfgang, The CLT has been discussed extensively on our site: browse the top-voted hits in a site search. It would be difficult to answer the question "what does a Gaussian Random Variable mean" without having a more specific understanding of "mean"--and for almost any conceivable interpretation, you are likely to find answers here due to the ubiquity of Gaussian (aka "Normal") distributions in the theory and practice of statistical analysis. $\endgroup$
    – whuber ♦
    Commented Jul 14, 2014 at 19:53

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