Rayleigh distribution

Rayleigh distribution
Rayleigh distribution
	; Probability density
	; Distribution function
Options
Carrier
Probability density
Distribution function
Expected value
Median
Fashion
Dispersion
Asymmetry coefficient
Excess ratio
Differential entropy
The generating function of moments
Characteristic function

Rayleigh distribution is a probability distribution of a random variable $\displaystyle X$ ${\ displaystyle \ displaystyle X}$ $\ displaystyle X$ with density

f(x;\sigma )={\frac {x}{\sigma ^{2}}}\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right),x\geqslant 0,\sigma >0,

{\ displaystyle f (x; \ sigma) = {\ frac {x} {\ sigma ^ {2}}} \ exp \ left (- {\ frac {x ^ {2}} {2 \ sigma ^ {2} }} \ right), x \ geqslant 0, \ sigma> 0,}

{\ displaystyle f (x; \ sigma) = {\ frac {x} {\ sigma ^ {2}}} \ exp \ left (- {\ frac {x ^ {2}} {2 \ sigma ^ {2} }} \ right), x \ geqslant 0, \ sigma> 0,}

Where $\displaystyle \sigma$ ${\ displaystyle \ displaystyle \ sigma}$ $\ displaystyle \ sigma$ - scale parameter. The corresponding distribution function has the form

{\mathsf {P}}(X\leqslant x)=\int \limits _{0}^{x}f(\xi )\,d\xi =1-\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right),x\geqslant 0.

{\ displaystyle {\ mathsf {P}} (X \ leqslant x) = \ int \ limits _ {0} ^ {x} f (\ xi) \, d \ xi = 1- \ exp \ left (- {\ frac {x ^ {2}} {2 \ sigma ^ {2}}} \ right), x \ geqslant 0.}

{\ mathsf P} (X \ leqslant x) = \ int \ limits _ {0} ^ {x} f (\ xi) \, d \ xi = 1- \ exp \ left (- {\ frac {x ^ { 2}} {2 \ sigma ^ {2}}} \ right), x \ geqslant 0.

Introduced for the first time in 1880 by John William Strett (Lord Rayleigh) in connection with the problem of adding harmonic oscillations to random phases.

Content

Application

In the tasks of gun sighting. If deviations from the target for two mutually perpendicular directions are normally distributed and uncorrelated, the coordinates of the target coincide with the origin, then designating the scatter along the axes as $X$ ${\ displaystyle X}$ and $Y$ ${\ displaystyle Y}$ , we obtain the expression for the miss value in the form $R={\sqrt {{{X}^{2}}+{{Y}^{2}}}}$ ${\ displaystyle R = {\ sqrt {{{X} ^ {2}} + {{Y} ^ {2}}}}}$ . In this case, the value $R$ ${\ displaystyle R}$ has a Rayleigh distribution.
In radio engineering to describe the amplitude fluctuations of a radio signal.
The density distribution of the radiation of a black body in frequency.

Relationship with other distributions

If a ${X}$ ${\ displaystyle {X}}$ and ${Y}$ ${\ displaystyle {Y}}$ - independent Gaussian random variables with zero mathematical expectation and the same variance ${{\sigma }^{2}}$ ${\ displaystyle {{\ sigma} ^ {2}}}$ then the random variable $Z={\sqrt {{{X}^{2}}+{{Y}^{2}}}}$ ${\ displaystyle Z = {\ sqrt {{{X} ^ {2}} + {{Y} ^ {2}}}}}$ has a Rayleigh distribution.
If independent Gaussian random variables ${X}$ ${\ displaystyle {X}}$ and ${Y}$ ${\ displaystyle {Y}}$ have nonzero mathematical expectations, in general, unequal, then the Rayleigh distribution passes into the Rice distribution .
The distribution density of the square of the Rayleigh value with ${\sigma =1}$ ${\ displaystyle {\ sigma = 1}}$ has a chi-square distribution with two degrees of freedom.

Literature

Perov, A.I. Statistical Theory of Radio Engineering Systems. - M .: Radio Engineering, 2003 .-- 400 p. - ISBN 5-93108-047-3 .

Rayleigh distribution

Content

Application

Relationship with other distributions

See also

Literature

More articles:

Rayleigh distribution
Probability density
Distribution function
Options	$\sigma >0$ ${\ displaystyle \ sigma> 0}$ $\ sigma> 0$
Carrier	$x\in [0;\infty )$ ${\ displaystyle x \ in [0; \ infty)}$ $x \ in [0; \ infty)$
Probability density	${\frac {x}{{\sigma }^{2}}}\exp \left(-{\frac {{x}^{2}}{2{{\sigma }^{2}}}}\right)$ ${\ displaystyle {\ frac {x} {{\ sigma} ^ {2}}} \ exp \ left (- {\ frac {{x} ^ {2}} {2 {{\ \ sigma} ^ {2}} }} \ right)}$ ${\ frac {x} {{{\ sigma} ^ {{2}}}} \ exp \ left (- {\ frac {{{x} ^ {{2}}}} {2 {{\ \ sigma} ^ {{2}}}}} \ right)$
Distribution function	$1-\exp \left({\frac {-x^{2}}{2\sigma ^{2}}}\right)$ ${\ displaystyle 1- \ exp \ left ({\ frac {-x ^ {2}} {2 \ sigma ^ {2}}} \ right)}$ $1- \ exp \ left ({\ frac {-x ^ {2}} {2 \ sigma ^ {2}}} \ right)$
Expected value	${\sqrt {\frac {\pi }{2}}}\sigma$ ${\ displaystyle {\ sqrt {\ frac {\ pi} {2}}} \ sigma}$ ${\ sqrt {{\ frac {\ pi} {2}}}} \ sigma$
Median	$\sigma {\sqrt {\ln(4)}}$ ${\ displaystyle \ sigma {\ sqrt {\ ln (4)}}}$ ${\ displaystyle \ sigma {\ sqrt {\ ln (4)}}}$
Fashion	$\sigma$ ${\ displaystyle \ sigma}$ $\ sigma$
Dispersion	$\left(2-\pi /2\right){{\sigma }^{2}}$ ${\ displaystyle \ left (2- \ pi / 2 \ right) {{\ sigma} ^ {2}}}$ $\ left (2- \ pi / 2 \ right) {{\ sigma} ^ {{2}}}$
Asymmetry coefficient	${\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}$ ${\ displaystyle {\ frac {2 {\ sqrt {\ pi}} (\ pi -3)} {(4- \ pi) ^ {3/2}}}}$ ${\ frac {2 {\ sqrt {\ pi}} (\ pi -3)} {(4- \ pi) ^ {{3/2}}}}$
Excess ratio	$-{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}$ ${\ displaystyle - {\ frac {6 \ pi ^ {2} -24 \ pi +16} {(4- \ pi) ^ {2}}}}$ $- {\ frac {6 \ pi ^ {2} -24 \ pi +16} {(4- \ pi) ^ {2}}}$
Differential entropy	$1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}$ ${\ displaystyle 1+ \ ln \ left ({\ frac {\ sigma} {\ sqrt {2}}} \ right) + {\ frac {\ gamma} {2}}}$ $1+ \ ln \ left ({\ frac {\ sigma} {{\ sqrt {2}}}} \ right) + {\ frac {\ gamma} {2}}$
The generating function of moments	$1+\sigma t\,e^{\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left({\textrm {erf}}\left({\frac {\sigma t}{\sqrt {2}}}\right)\!+\!1\right)$ ${\ displaystyle 1+ \ sigma t \, e ^ {\ sigma ^ {2} t ^ {2} / 2} {\ sqrt {\ frac {\ pi} {2}}} \ left ({\ textrm {erf }} \ left ({\ frac {\ sigma t} {\ sqrt {2}}} \ right) \! + \! 1 \ right)}$ $1+ \ sigma t \, e ^ {{\ sigma ^ {2} t ^ {2} / 2}} {\ sqrt {{\ frac {\ pi} {2}}}} \ left ({\ textrm { erf}} \ left ({\ frac {\ sigma t} {{\ sqrt {2}}}} \ right) \! + \! 1 \ right)$
Characteristic function	$1\!-\!\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\!\left({\textrm {erfi}}\!\left({\frac {\sigma t}{\sqrt {2}}}\right)\!-\!i\right)$ ${\ displaystyle 1 \! - \! \ sigma te ^ {- \ sigma ^ {2} t ^ {2} / 2} {\ sqrt {\ frac {\ pi} {2}}} \! left ({ \ textrm {erfi}} \! \ left ({\ frac {\ sigma t} {\ sqrt {2}}} \ right) \! - \! i \ right)}$ $1 \! - \! \ Sigma te ^ {{- \ sigma ^ {2} t ^ {2} / 2}} {\ sqrt {{\ frac {\ pi} {2}}}}!! Left ( {\ textrm {erfi}} \! \ left ({\ frac {\ sigma t} {{\ sqrt {2}}}} \ right) \! - \! i \ right)$