Rayleigh mixture distribution

In probability theory and statistics a Rayleigh mixture distribution is a weighted mixture of multiple probability distributions where the weightings are equal to the weightings of a Rayleigh distribution. Since the probability density function for a (standard) Rayleigh distribution is given by f ( x ; σ ) = x σ 2 e − x 2 / 2 σ 2 , x ≥ 0 , {\displaystyle f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/2\sigma ^{2}},\quad x\geq 0,} Rayleigh mixture distributions have probability density functions of the form f ( x ; σ , n ) = ∫ 0 ∞ r e − r 2 / 2 σ 2 σ 2 τ ( x , r ; n ) d r , {\displaystyle f(x;\sigma ,n)=\int _{0}^{\infty }{\frac {re^{-r^{2}/2\sigma ^{2}}}{\sigma ^{2}}}\tau (x,r;n)\,\mathrm {d} r,} where τ ( x , r ; n ) {\displaystyle \tau (x,r;n)} is a well-defined probability density function or sampling distribution.

Source: Wikipedia — Rayleigh mixture distribution (CC BY-SA 4.0)

Rayleigh mixture distribution

In probability theory and statistics a Rayleigh mixture distribution is a weighted mixture of multiple probability distributions where the weightings are equal to the weightings of a Rayleigh distribution. Since the probability density function for a (standard) Rayleigh distribution is given by f ( x ; σ ) = x σ 2 e − x 2 / 2 σ 2 , x ≥ 0 , {\displaystyle f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/2\sigma ^{2}},\quad x\geq 0,} Rayleigh mixture distributions have probability density functions of the form f ( x ; σ , n ) = ∫ 0 ∞ r e − r 2 / 2 σ 2 σ 2 τ ( x , r ; n ) d r , {\displaystyle f(x;\sigma ,n)=\int _{0}^{\infty }{\frac {re^{-r^{2}/2\sigma ^{2}}}{\sigma ^{2}}}\tau (x,r;n)\,\mathrm {d} r,} where τ ( x , r ; n ) {\displaystyle \tau (x,r;n)} is a well-defined probability density function or sampling distribution.

Source: Wikipedia "Rayleigh mixture distribution" · CC BY-SA 4.0

Share this article: X · Bluesky
Privacy Policy