Wrapped Lévy distribution

In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle.

Description

The pdf of the wrapped Lévy distribution is

f_{WL}(\theta ;\mu ,c)=\sum _{n=-\infty }^{\infty }{\sqrt {\frac {c}{2\pi }}}\,{\frac {e^{-c/2(\theta +2\pi n-\mu )}}{(\theta +2\pi n-\mu )^{3/2}}}

where the value of the summand is taken to be zero when $\theta +2\pi n-\mu \leq 0$ , $c$ is the scale factor and $\mu$ is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:

f_{WL}(\theta ;\mu ,c)={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{-in(\theta -\mu )-{\sqrt {c|n|}}\,(1-i\operatorname {sgn} {n})}={\frac {1}{2\pi }}\left(1+2\sum _{n=1}^{\infty }e^{-{\sqrt {cn}}}\cos \left(n(\theta -\mu )-{\sqrt {cn}}\,\right)\right)

In terms of the circular variable $z=e^{i\theta }$ the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:

\langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{WL}(\theta ;\mu ,c)\,d\theta =e^{in\mu -{\sqrt {c|n|}}\,(1-i\operatorname {sgn}(n))}.

where $\Gamma \,$ is some interval of length $2\pi$ . The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:

\langle z\rangle =e^{i\mu -{\sqrt {c}}(1-i)}

The mean angle is

\theta _{\mu }=\mathrm {Arg} \langle z\rangle =\mu +{\sqrt {c}}

and the length of the mean resultant is

R=|\langle z\rangle |=e^{-{\sqrt {c}}}

References

Fisher, N. I. (1996). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 978-0-521-56890-6. Retrieved 2010-02-09.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
Multinomial
- Dirichlet
- Negative
Continuous:
Dirichlet
- Generalized
Multivariate Laplace
Multivariate normal
Multivariate stable
Multivariate t
Normal-gamma
- Inverse
Matrix-valued:
LKJ
Matrix beta
Matrix normal
Matrix t
Matrix gamma
- Inverse
Wishart
- Normal
- Inverse
- Normal-inverse
- Complex

Directional

Univariate (circular) directional: Circular uniform; Univariate von Mises; Wrapped normal; Wrapped Cauchy; Wrapped exponential; Wrapped asymmetric Laplace; Wrapped Lévy
Bivariate (spherical): Kent
Bivariate (toroidal): Bivariate von Mises
Multivariate: von Mises–Fisher; Bingham

Degenerate
and singular

Degenerate: Dirac delta function
Singular: Cantor

Families

Read other articles:

Koto Damai, Kampar Kiri Tengah, Kampar

Koordinat: 0°13′13.7″N 101°14′07.4″E / 0.220472°N 101.235389°E / 0.220472; 101.235389Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Koto Damai, Kampar Kiri Tengah, Kampar – berita · surat kabar · buku · cendekiawan ...

Kafana

Sebuah kafana di Borač, Distrik Šumadija, Serbia. Kafana (dalam bahasa Bosnia, Montenegro, dan Serbia pelafalan [kafǎna]), kafeana (кафеана, dalam bahasa Makedonia), kavana (dalam bahasa Kroasia, IPA: [kaʋǎna]) adalah istilah yang mengacu kepada sejenis bistro lokal di negara-negara bekas Yugoslavia yang menyajikan minuman beralkohol dan kopi serta makanan-makanan ringan (meze) atau makanan lain. Banyak pula kafana yang mengadakan pertunjukan musik langsung. Konsep...

Richard Brooks

Untuk orang lain dengan nama yang sama, lihat Richard Brooks (disambiguasi). Richard BrooksBrooks pada 1986LahirReuben Sax18 Mei 1912Philadelphia, Pennsylvania, ASMeninggal11 Maret 1992(1992-03-11) (umur 79)Studio City, Los Angeles, California, ASSuami/istriJean Brooks (m. 1941; bercerai 1944) Harriette Levin (m. 1946; bercerai 1957) Jean Simmons (m. 1960...

Bagan Melibur, Merbau, Kepulauan Meranti

Bagan MeliburDesaKantor Desa Bagan MeliburNegara IndonesiaProvinsiRiauKabupatenKepulauan MerantiKecamatanMerbauKode pos28752Kode Kemendagri14.10.05.2004 Luas4107 HaJumlah penduduk2.235 jiwa Bagan Melibur merupakan salah satu desa yang berada di kecamatan Merbau, Kabupaten Kepulauan Meranti, provinsi Riau, Indonesia. Desa Bagan Melibur terletak diantara Kelurahan Teluk Belitung dan Desa Mayang Sari. Sejarah Sebelum kemerdekaan Indonesia, tahun 1942 Desa Bagan Melibur adalah sebuah kampung...

That's My Wife (1929 film)

1929 short film by Lloyd French Not to be confused with British quota quickie That's My Wife (1933 film). That's My WifeTheatrical release posterDirected byLloyd FrenchWritten byLeo McCarey (story)H. M. Walker (titles)Produced byHal RoachStarringStan LaurelOliver HardyCinematographyGeorge StevensEdited byRichard C. CurrierDistributed byMetro-Goldwyn-MayerRelease date March 23, 1929 (1929-03-23) Running time20 minutesCountryUnited StatesLanguagesSilent filmEnglish (Original inte...

WVNS-TV

CBS/Fox/MyNetworkTV affiliate in Lewisburg, West Virginia WVNS redirects here. For the Nashville, Tennessee, radio station which once used the WVNS-FM callsign, see WPRT-FM. WVNS-TVLewisburg–Bluefield–Beckley, West VirginiaUnited StatesCityLewisburg, West VirginiaChannelsDigital: 11 (VHF)Virtual: 59BrandingCBS 59; Fox 5959 NewsProgrammingAffiliations59.1: CBS59.2: Fox/MyNetworkTVOwnershipOwnerNexstar Media Group(Nexstar Media Inc.)HistoryFoundedJune 18, 1992First air dateAugust 12, 1...

Tyrrhenians

Name used by the ancient Greeks to refer to non-Greek people This article is about the Greek exonym for certain non-Greek peoples. For other uses, see Tyrrhenian (disambiguation). Tyrrhenians (Attic Greek: Τυῤῥηνοί Turrhēnoi) or Tyrsenians (Ionic: Τυρσηνοί Tursēnoi; Doric: Τυρσανοί Tursānoi[1]) was the name used by the ancient Greeks authors to refer, in a generic sense, to non-Greek people, in particular pirates.[2] While ancient sources have been...

يحيى بن معين

يحيى بن ‌مَعِين تخطيط يحيى بن معين بخط الثُّلث ملحوقًا بالترضي عنه: «» معلومات شخصية الميلاد سنة 775 [1] بغداد الوفاة 233 هـبغداد مواطنة الدولة العباسية المذهب الفقهي أهل السنة والجماعة الحياة العملية العصر القرن الثاني للهجرة والقرن الثالث للهجرة المنطقة بغدا...

Космические силы США

Космические силы СШАангл. United States Space Force Эмблема Космических сил США Годы существования 20 декабря 2019 — н. в. Страна США Подчинение Военно-воздушное министерство США Входит в Вооружённые силы США Тип вид вооружённых сил Включает в себя Командование космических о...

President of Croatia

Head of state and commander-in-chief of Croatia President of the Republic of CroatiaPredsjednik Republike HrvatskeCoat of arms of the presidentStandard of the presidentIncumbentZoran Milanovićsince 19 February 2020Office of the President of the RepublicStyleMr President (informal)His Excellency (diplomatic)TypeHead of stateMember ofArmed Forces of CroatiaReports toCroatian ParliamentResidence36 Krajiška Street, Zagreb[1][2]SeatPredsjednički dvori, ZagrebAppointerPopula...

Theodore Roosevelt III

American governmental official This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Theodore Roosevelt III – news · newspapers · books · scholar · JSTOR (March 2014) (Learn how and when to remove this message) Theodore Roosevelt IIISecretary of Commerce of PennsylvaniaIn office1949–1951 Personal detailsBornTheo...

Elena Berezovich

Russian linguist This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (March 2018) (Learn how and when to remove this message) Elena Berezovich Elena Lvovna Berezovich (Russian: Елена Львовна Березович, born 1966) is a Russian linguist known for her work in onomastics, etymology, and ethnolinguistics. She is c...

Rumpun bahasa Jermanik Utara

Cari artikel bahasa Cari berdasarkan kode ISO 639 (Uji coba) Cari berdasarkan nilai Glottolog Kolom pencarian ini hanya didukung oleh beberapa antarmuka Halaman rumpun acak Rumpun bahasaJermanik Utara NordikSkandinaviaPersebaranEropa UtaraPenggolongan bahasaIndo-EropaJermanikJermanik Utara Nordik Greenland Islandia Faeroer Norn Norwegia Denmark Swedia Gutniska Bahasa indukProto-NordikKode bahasaISO 639-5gmqGlottolognort3160 Portal BahasaSunting kotak info • L&...

Ryan Giggs

Welsh footballer (born 1973) Ryan GiggsOBE Giggs with Manchester United in 2015Personal informationFull name Ryan Joseph Giggs[1]Birth name Ryan Joseph WilsonDate of birth (1973-11-29) 29 November 1973 (age 50)Place of birth Canton, Cardiff, WalesHeight 5 ft 10 in (1.79 m)[2]Position(s) Wide midfielderYouth career Deans FC1985–1987 Manchester City1987–1990 Manchester UnitedSenior career*Years Team Apps (Gls)1990–2014 Manchester United 672 (114)Interna...

Mountain View, California

City in California, United States For other places in California named Mountain View, see Mountain View (disambiguation). City in California, United StatesMountain View, CaliforniaCityCity Hall & Performing Arts Center FlagSealLocation within Santa Clara CountyMountain View city map, California, U.S.Coordinates: 37°23′10″N 122°05′02″W / 37.38611°N 122.08389°W / 37.38611; -122.08389CountryUnited StatesStateCaliforniaCountySanta ClaraIncorporatedNovember ...

Save the Children

Non-government organization founded 1919 For other uses, see Save the Children (disambiguation). Not to be confused with Save Our Children, Save the Children USA, or Save the Kids token. This article contains wording that promotes the subject in a subjective manner without imparting real information. Please remove or replace such wording and instead of making proclamations about a subject's importance, use facts and attribution to demonstrate that importance. (March 2023) (Learn how and when ...

Carl Froelich

Carl Froelich durante le riprese del film Die Regimentstochter (1908) accanto a Oskar Messter Carl Froelich, nato Carl August Hugo Froelich, (Berlino, 5 settembre 1875 – Berlino, 12 febbraio 1953), è stato un regista, direttore della fotografia, produttore cinematografico e sceneggiatore tedesco. È considerato uno dei pionieri del cinema tedesco. Indice 1 Filmografia 1.1 Regista 1.2 Produttore 1.3 Sceneggiatore 1.4 Direttore della fotogragia 2 Altri progetti 3 Collegamenti esterni Filmogr...

The Storm at Sea

Painting by Pieter Bruegel the Elder Not to be confused with Stormy Sea. Storm at SeaArtistPieter Bruegel the ElderYear1569TypeOil on panelDimensions70.3 cm × 97 cm (27.7 in × 38 in)LocationKunsthistorisches Museum, Vienna Storm at Sea is an oil painting on panel by the Netherlandish Renaissance artist Pieter Bruegel the Elder, painted in c. 1569. It is in the Kunsthistorisches Museum in Vienna. Description In the past doubts have been raised about ...

Shigeru Yoshida

Prime Minister of Japan (1878–1967) Shigeru Yoshida吉田茂Prime Minister of JapanIn office15 October 1948 – 10 December 1954MonarchHirohitoGovernorDouglas MacArthurMatthew RidgwayDeputyJōji Hayashi [jp]Taketora OgataPreceded byHitoshi AshidaSucceeded byIchirō HatoyamaIn office22 May 1946 – 24 May 1947MonarchHirohitoGovernorDouglas MacArthurDeputyKijūrō ShideharaPreceded byKijūrō ShideharaSucceeded byTetsu KatayamaMinister of Foreign AffairsIn offi...

Printf (Unix)

Standard UNIX utility This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Printf Unix – news · newspapers · books · scholar · JSTOR (July 2010) (Learn how and when to remove this message) printfDeveloper(s)Various open-source and commercial developersOperating systemUnix and Unix-likePlatformCross-platformT...

PROFILPELAJAR.COM