Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. It is essentially a chi distribution with two degrees of freedom. In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. It is essentially a chi distribution with two degrees of freedom. A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions.Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed. The distribution is named after Lord Rayleigh (/ˈreɪli/). The probability density function of the Rayleigh distribution is where σ {displaystyle sigma } is the scale parameter of the distribution. The cumulative distribution function is for x ∈ [ 0 , ∞ ) . {displaystyle xin [0,infty ).} Consider the two-dimensional vector Y = ( U , V ) {displaystyle Y=(U,V)} which has components that are normally distributed, centered at zero, and independent. Then U {displaystyle U} and V {displaystyle V} have density functions Let X {displaystyle X} be the length of Y {displaystyle Y} . That is, X = U 2 + V 2 . {displaystyle X={sqrt {U^{2}+V^{2}}}.} Then X {displaystyle X} has cumulative distribution function where D x {displaystyle D_{x}} is the disk