Statistics - Rayleigh Distribution

# Statistics – Rayleigh Distribution

The Rayleigh distribution is a distribution of steady chance density operate. It’s named after the English Lord Rayleigh. This distribution is broadly used for the next:

• CommunicationsÂ – to mannequin a number of paths of densely scattered indicators whereas reaching a receiver.
• Bodily SciencesÂ – to mannequin wind pace, wave heights, sound or mild radiation.
• EngineeringÂ – to examine the lifetime of an object relying upon its age.
• Medical ImagingÂ – to mannequin noise variance in magnetic resonance imaging.

The chance density operate Rayleigh distribution is outlined as:

## Formulation

f(x;Ïƒ)=xÏƒ2eâˆ’x22Ïƒ2,xâ‰¥0f(x;Ïƒ)=xÏƒ2eâˆ’x22Ïƒ2,xâ‰¥0

The place âˆ’

• ÏƒÏƒÂ = scale parameter of the distribution.

The comulative distribution operate Rayleigh distribution is outlined as:

## Formulation

F(x;Ïƒ)=1âˆ’eâˆ’x22Ïƒ2,xâˆˆ[0âˆžF(x;Ïƒ)=1âˆ’eâˆ’x22Ïƒ2,xâˆˆ[0âˆž

Where âˆ’

• ÏƒÏƒÂ = scale parameter of the distribution.

## Variance and Expected Value

The expected value or the mean of a Rayleigh distribution is given by:

E[x]=ÏƒÏ€2âˆ’âˆ’âˆšE[x]=ÏƒÏ€2

The variance of a Rayleigh distribution is given by:

Var[x]=Ïƒ24âˆ’Ï€2