Thread: Rayleigh pdf ratio and parameter estimation

1. Rayleigh pdf ratio and parameter estimation

Dear all,
I've a problem regarding Rayleigh pdf.
In particular, I know the ratio $\displaystyle X$ between two part of the probability density function, at the right and at the left of a given threshold point $\displaystyle th$, so

$\displaystyle \int_{th}^{\infty}f(x,\sigma)dx=X\int_{0}^{th}f(x, \sigma)dx$

and from the former I need to compute the value of the parameter $\displaystyle \sigma$ of the Rayleigh distribution $\displaystyle f(x,\sigma)$.

I've made the first steps, reducing the former in

$\displaystyle \int_{0}^{th}f(x,\sigma)dx=\frac{1}{1+X}$

and solving the definite integral, obtaining

$\displaystyle \frac{\frac{1}{\sigma^{2}}\left( e^{\frac{-th^{2}}{2\sigma^{2}}}-1\right) }{1-\frac{1}{\sigma^{2}}}=\frac{1}{1+X}$

However, is now possible to solve the last equation in $\displaystyle \sigma$? otherwise, is there any other method to compute (or, at least, estimate) the value of $\displaystyle \sigma$ from the first equation?

Thank you in advance for any suggestion!
Simo

2. Hello,

The cumulative density function of a Rayleigh distribution is (according to the wikipedia) :

$\displaystyle \int_0^t f(x,\sigma) ~dx=1-\exp\left(\frac{-t^2}{2\sigma^2}\right)$

So you actually have :

$\displaystyle 1-\exp\left(\frac{-th^2}{2\sigma^2}\right)=\frac{1}{1+X}$

But it looks like you don't have the same definition of the Rayleigh distribution than the wikipedia...