title: Jim Gaffigan

text: As Gaffigan's comedy career stalled in the 1990s, a friend suggested he audition for commercials, a move that turned out to be profitable. He has appeared in over 200 TV commercials, ranging from Rolling Rock to Saturn to Chrysler and ESPN. His ubiquity earned him the title of 'Salesman of the Year' by "BusinessWeek" in 1999. He also performed in a trio of Sierra Mist commercials for the 2007 Super Bowl as part of the Sierra Mist comedy ensemble "The Mis-Takes". He appeared in an ad series for Sierra Mist alongside fellow comedian Michael Ian Black.


title: Jim Gaffigan

text: After his first appearance on "The Late Show with David Letterman" in 1999, Gaffigan was tapped by the host to develop a sitcom called "Welcome to New York" in which he also co-starred alongside Christine Baranski. The show was cancelled after its first season despite receiving positive reviews. During the 2000/2001 television season, he was a cast member of "The Ellen Show" on CBS, Ellen DeGeneres's second sitcom. He appeared in two movies chosen for the 2001 Sundance Film Festival: "Super Troopers" and "30 Years to Life". He appeared on "That '70s Show". He was a regular cast member of the TBS original sitcom "My Boys". He left the show at the end of the third season.


title: Jim Gaffigan

text: In 2009, Gaffigan guest starred as the best friend of Murray Hewitt in one episode of the HBO comedy series "Flight of the Conchords". Later that same year, he appeared in the Sam Mendes-directed dramedy "Away We Go" and the teen comedy "17 Again". On June 11, 2009, Gaffigan appeared on "The Tonight Show with Conan O'Brien". He appeared on "Law & Order" episodes "Flight" and "Reality Bites", as well as an episode titled "Smile" on "Law and Order: Criminal Intent".


title: Jim Gaffigan

text: He was in an episode of "The Daily Show" as a man posing as a "Daily Show" correspondent who knows nothing about the show (he refers to it as "The John Daily Show") and simply wants to be seen with Jon Stewart. This was meant to be a parody of the 2009 White House gatecrash incident.


title: Jim Gaffigan

text: Gaffigan appeared on Broadway in "That Championship Season", which opened in March 2011, opposite Brian Cox, Chris Noth, Kiefer Sutherland, and Jason Patric. Gaffigan's performance was praised by ABC News correspondent Sandy Kenyon as the most moving and that he may "steal the show". He called being on Broadway "an amazing experience, really hard but really fun".


title: Jim Gaffigan

text: Gaffigan starred in Shia LaBeouf's 2013 short film "Howard Cantour.com", the content of which was later revealed to be mostly plagiarized from Daniel Clowes's 2007 graphic novella "Justin M. Damiano". Reflecting on the incident in an interview for "The Daily Beast", Gaffigan said, "There's no greater sin in the stand-up world than thievery...So you do not want to be associated with thievery," but added, "I don't have any hard feelings about it because I don't think people think I had anything to do with it."


title: Jim Gaffigan

text: In the 2010s, Gaffigan, his wife, and Peter Tolan began to develop material for a show based loosely on their own life. CBS agreed to shoot a pilot of their show in March 2013, with casting by Marc Hirschfeld, and Mira Sorvino playing his wife; but ultimately passed on the project. When the cable network TV Land began efforts to broadcast original material and attract younger audiences, it offered the Gaffigans complete creative control. The result was "The Jim Gaffigan Show", a sitcom about a couple raising their five kids in a two-bedroom New York City apartment. After the release of two online-only episodes in June 2015, the pilot episode aired on July 15, 2015. The show stars Gaffigan as a fictionalized version of himself, with his wife Jeannie played by Ashley Williams. Other characters include their real-estate agent (and Jeannie's best friend) Daniel (played by Michael Ian Black), Jim's fellow comic and best friend Dave (Adam Goldberg), and their priest, Father Nicholas (Tongayi Chirisa). After 2 seasons of the show, in 2016 Jim and his wife Jeannie decided not to continue with a third season so they could spend more time with their kids.


title: Jim Gaffigan

text: Gaffigan co-starred in the film "Experimenter", a fictionalized account of the experiments of the Yale professor Stanley Milgram. Gaffigan plays an actor hired to collaborate in the experiments.


title: Jim Gaffigan

text: In February 2016, Gaffigan began appearing in KFC commercials as Colonel Sanders, replacing Norm Macdonald.


title: Jim Gaffigan

text: Since April 8, 2016, Gaffigan has appeared with his family in a marketing campaign for the 2017 Chrysler Pacifica.


title: Jim Gaffigan

text: In October 2016, it was announced Gaffigan would be joining the cast of the third season of the anthology drama series, "Fargo". However, he was ultimately forced to drop out due to scheduling difficulties; he was replaced by Mark Forward. In 2018, he portrayed Paul Markham in the biographical drama "Chappaquiddick", starring Jason Clarke and Kate Mara, to positive reviews. Gaffigan next took on the lead role in the 2018 neo-noir film "American Dreamer", which was released on September 20, 2019. Gaffigan's dark portrayal of a broken man driven to desperate actions was very well received by critics, with the Chicago Sun-Times review calling it "a career-best dramatic performance" by Gaffigan.


title: Jim Gaffigan

text: In 2020, Gaffigan was announced to provide the voice of Thunderbolt in season two of "Stargirl". By season three, the role recast to Seth Green.


title: Jim Gaffigan

text: In January 2021, he joined the cast of Disney's "Peter Pan & Wendy" as Mr. Smee. In 2021, Gaffigan voiced Lorenzo Paguro in the Pixar film "Luca".


title: Jim Gaffigan

text: Gaffigan is also notable for his voice-over work. He voiced an animated version of himself on "Pale Force" with Conan O'Brien from 2005 to 2008. He has also voiced characters on the animated shows "Bob's Burgers", "Shorty McShorts' Shorts", "WordGirl", "Star vs. the Forces of Evil", and the animated feature "Duck Duck Goose".


title: Jim Gaffigan

text: Gaffigan produced a series of animated shorts for "Late Night with Conan O'Brien", titled "Pale Force" (2005–2008). The animated sketches featured Gaffigan and O'Brien as superheroes who fight crime with their extremely pale skin. The series was nominated for a Daytime Emmy in 2007 in the category of "Outstanding Broadband Program – Comedy".


title: Jim Gaffigan

text: Gaffigan's humorous quips have earned him over two and a half million followers on Twitter. He was listed by "Rolling Stone" as one of the "25 funniest people on Twitter" in 2012.


title: Jim Gaffigan

text: In 2013, Gaffigan released "Dad Is Fat", a title derived from the first complete sentence his eldest son wrote on a dry-erase board at the age of four or five. "He showed it to me," Gaffigan recalled in an interview, "and I laughed, and then I put him up for adoption." The book is a collection of essays dealing with the raising of his children, as well as reminiscences from his own childhood. In support of the volume, he appeared on NPR's "Weekend Edition", ABC's "The View", and MSNBC's "Morning Joe", spoke at BEA in New York, embarked on a nine-stop bus tour that ended on Father's Day.


title: Jim Gaffigan

text: It debuted at number five on "The New York Times" Best Seller's list, remaining on the list for three months. The book received tepid reviews from critics. "Kirkus Reviews" said of the book that it's "hardly groundbreaking comedy material, but the book will appeal to Gaffigan's fans". Lou Harry of the Indianapolis Business Journal said that while "no new ground is broken in Jim Gaffigan's book...'Dad is Fat' should be a fun intermezzo in your summer reading pile." Regarding the audiobook, which Gaffigan read, "Audiofile" said his "performance strikes the right balance between his near-deadpan comedy delivery and the energy needed to keep a beleaguered parent engaged".


title: Jim Gaffigan

text: Gaffigan signed with Crown Publishing in June 2013 to write a second book of comic essays. The book, "Food: A Love Story", which was released in Fall 2014. "Publishers Weekly" said the book "packs plenty of laughs". "Kirkus Reviews" remarked that "Gaffigan somehow manages to work 'clean' without ever becoming sickeningly saccharine," and that "laughs [are] served up just right on every page". Of the accompanying audiobook, the "Library Journal" said, "The witty commentary is peppered with jokes and funny stories that will have listeners smiling throughout and occasionally laughing out loud."


title: Jim Gaffigan

text: Prior to meeting his wife, actress Jeannie Noth, Gaffigan largely wrote alone. However, while working on his first show, "Welcome to New York", he was overwhelmed and asked for input from her (then his friend). Although initially hesitant to have a collaborator, as their relationship grew, so did Noth's ability to write material for him. Once they married, she left behind her work with her youth theater project (Shakespeare on the Playground) to devote herself to raising their expanding brood, and after a joke she wrote drew big laughs at a show, she and Jim began to collaborate more. She gradually transferred into the position of Jim's chief co-writer, and they are now full writing partners. She has been a credited writer and/or executive producer on all his comedy endeavors since "Beyond the Pale", including his two books and television show. He also credits her with "coaching" him through his performance in "That Championship Season".


title: Jim Gaffigan

text: Gaffigan participated on the NPR radio quiz program "Wait Wait... Don't Tell Me!" in 2013. Gaffigan is also a regular commentator on "CBS Sunday Morning", for which he won a Daytime Emmy in 2016.


title: Jim Gaffigan

text: Gaffigan married actress Jeannie Gaffigan (née Noth) in 2003. They have two daughters and three sons together. The family of seven lived in a two-bedroom apartment in the Manhattan borough of New York City, before moving to a larger Manhattan home in 2015. To stay connected to his family, Gaffigan tries to "maintain bedtime rituals while working in the city". When on tour, he reportedly takes his family with him.


title: Jim Gaffigan

text: Gaffigan is a practicing Catholic who avoids working on Sundays, though he has joked that his wife is so much more devout than him that she seems like a "Shiite Catholic" in comparison. He and his family attend Mass at St. Patrick's Old Cathedral in Manhattan, which is also where he and his wife were married and where their children were baptized. In May 2016, he and his wife delivered the commencement address at the Catholic University of America. They also delivered the commencement address at his wife's alma mater, Marquette University, in May 2018. They were both awarded an honorary Doctor of Humane Letters degree as part of the graduation ceremony.


title: Jim Gaffigan

text: On April 26, 2014, Gaffigan received the award for Concert Comedian at the American Comedy Awards for his work.


title: Jim Gaffigan

text: In 2007, "Pale Force" was nominated for a Daytime Emmy for Outstanding Broadband Program in the Comedy category. Gaffigan served as executive producer, writer, and lead actor.


title: Jim Gaffigan

text: In 2016, he won the Daytime Emmy Award for Outstanding Morning Program for his contributions as a commentator to "CBS Sunday Morning".


title: Jim Gaffigan

text: The Grammy Awards are awarded annually by the National Academy of Recording Arts and Sciences. Gaffigan has been nominated seven times.


title: Shehnaaz Gill

text: Shehnaaz Gill (born 27 January 1993 or 1994), also known as Shehnaaz Kaur Gill, is an Indian actress, model and singer who primarily works in Hindi television and films. She began her modelling career with the 2015 music video "Shiv Di Kitaab". In 2017, she debuted as an actress in the Punjabi film "Sat Shri Akaal England". In 2019, Gill participated in the reality show "Bigg Boss 13", where she finished in third place.


title: Shehnaaz Gill

text: Gill was born on 27 January 1993 or 1994 and raised in Punjab, India. She is of Punjabi descent and belongs to a Sikh family. She loved to act and sing, and dreamt to be an actor since childhood.


title: Shehnaaz Gill

text: She did her schooling from Dalhousie Hilltop School, Dalhousie and completed her graduation from Lovely Professional University, Phagwara, and has a degree in commerce.


title: Shehnaaz Gill

text: Gill began her career by featuring in "Shiv Di Kitaab" in 2015. Gill appeared in "Majhe Di Jatti" and "Pindaan Diyan Kudiyaan" in 2016. Gill's another music video came with Garry Sandhu titled Yeah Baby Refix. Gill also starred in some Punjabi films such as "Sat Shri Akaal England" in 2017, "Kala Shah Kala" and "Daaka" in 2019.


title: Shehnaaz Gill

text: In September 2019, Gill entered as a celebrity participant on "Bigg Boss 13". While she was in the "Bigg Boss" house, her first single, "Veham" came out, followed by some other singles including "Sidewalk", "Range" and "Ronda Ali Peti". The season ended in February 2020, where Gill finished up as second runner-up. In February 2020, she appeared in "Mujhse Shaadi Karoge" but it terminated within a month because of the COVID-19 pandemic.


title: Shehnaaz Gill

text: Gill then appeared in a number of music videos including "Bhula Dunga", "Keh Gayi Sorry", "Kurta Pajama", "Waada Hai", "Shona Shona" and "Fly".


title: Debbie Gibson

text: Deborah Ann Gibson (born August 31, 1970) is an American singer-songwriter, record producer and actress.


title: Debbie Gibson

text: Gibson released her debut album "Out of the Blue" in 1987, which spawned several international hits, later being certified triple platinum by the Recording Industry Association of America. One of those singles, "Foolish Beat", made Gibson the youngest female artist to write, produce, and perform a "Billboard" Hot 100 number-one single. Her double-platinum second album "Electric Youth" (1989), gave Gibson another U.S. number-one hit with "Lost in Your Eyes". Gibson is the sole songwriter on all of her singles to reach the top 20 of the Billboard Hot 100. She was recognized by ASCAP as Songwriter of the Year, along with Bruce Springsteen, in 1989.


title: Debbie Gibson

text: Gibson continued to record and release music throughout the 1990s and 2000s. In 2006, she reached number 24 on the U.S. adult contemporary chart with "Say Goodbye", a duet with Jordan Knight, and in 2017 achieved her highest-charting hit in more than 25 years in her duet with Sir Ivan, "I Am Peaceman". Gibson's 2020 single "Girls Night Out" became her first top five and highest-charting hit after 30 years. In addition to music, she has gone on to starring roles on Broadway and touring musicals, including playing Eponine in "Les Misérables" and Sandy in "Grease", as well as television and independent film work.


title: Debbie Gibson

text: Gibson was born on Long Island, New York, the third of Diane (née Pustizzi) and Joseph Gibson's four daughters. Her father, who enjoyed singing, was originally named Joseph Schultz and was orphaned as a boy; his biological mother married a man with the surname Gibson before putting Joseph into an orphanage. Gibson grew up in suburban Merrick, New York, on Long Island. She describes herself as of "Italian/Sicilian and part German and possibly some Russian" descent. She studied piano under American pianist Morton Estrin.


title: Debbie Gibson

text: In 1983, Gibson submitted a cassette recording of her original composition "I Come From America" to WOR for the station's song contest. After she won a cash prize of 1,000 from the contest, her mother convinced a relative to loan her 10,000 to convert the family garage into a recording studio.


title: Debbie Gibson

text: After years of writing and producing her own material, a demonstration tape of Gibson's submitted to a radio personality was eventually shared with an executive at Atlantic Records. Based solely on Gibson's original song, "Only in My Dreams", she was signed to a development deal and began a promotional tour of club venues throughout the United States.


title: Debbie Gibson

text: Gibson spent much of 1986 and the beginning of 1987 building her songwriting catalog, while continuing to play club dates. During her promotional tour, Gibson continued attending classes at Calhoun High School in Merrick, New York, where she later graduated as an honor student. Diane Gibson, Debbie's mother and manager, accompanied her daughter on many of these track dates. "We played dance clubs, straight clubs, and gay clubs," Diane has said.


title: Debbie Gibson

text: The "Only in My Dreams" single release landed the song within the "Billboard" Hot 100 reaching a peak of number 4 on that chart. Atlantic Records and Gibson began the process of completing the first album as a result of the good showing by the single. Along with producer Fred Zarr, Gibson wrote, recorded, and produced her first album "Out of the Blue" in only four weeks.


title: Debbie Gibson

text: Following the success of "Only In My Dreams", "Shake Your Love" was released as the follow-up single and reached the "Billboard" top five. The "Shake Your Love" video was choreographed by Paula Abdul and was the first time MTV had visited Gibson on a video shoot.


title: Debbie Gibson

text: In 1987, while performing at nightclub venues throughout the United States, Gibson was recording her debut album, "Out of the Blue". It was finished within four weeks. Four singles from the album reached the top five of the Hot 100 chart: "Only in My Dreams", "Shake Your Love", "Out of the Blue", and the number-one hit "Foolish Beat", followed by "Staying Together", which performed more modestly, reaching number 22. "Foolish Beat" set a record for Gibson, making her (at 16) the youngest artist ever to have written, produced, and performed a "Billboard" number-one single, as entered in the 1988 "Guinness Book of World Records". She remains the youngest female artist to have done so. "Out of the Blue" was established as a hit album, and she had success in the UK and southeast Asia, filling stadiums with her Out of the Blue tour. By the end of 1988, "Out of the Blue" had gone triple platinum.


title: Debbie Gibson

text: The music video compilation "Out of the Blue" was certified platinum by the RIAA; the concert tour video was certified double platinum. In October 1988, Gibson sang the national anthem for Game 1 of the Major League Baseball World Series.


title: Debbie Gibson

text: "Electric Youth" was released in early 1989 and spent five weeks at number one on the Billboard 200. The first single released, "Lost in Your Eyes", was number one on the Hot 100 for three weeks, garnering her another achievement as the youngest female to have both an album and single simultaneously at number one. She shared the 1989 ASCAP Songwriter of the Year Award with Bruce Springsteen. Subsequent singles from the album were "Electric Youth" (number 11), "No More Rhyme" (number 17), and "We Could Be Together" (number 71). The "Electric Youth" album was certified double platinum by the RIAA. The successful Electric Youth world tour and "Live Around the World" VHS (double platinum) followed.


title: Debbie Gibson

text: She recorded two more albums for Atlantic Records: "Anything Is Possible" (1990) and "Body, Mind, Soul" (1993). The former's title song, co-written with Motown mainstay Lamont Dozier, peaked at number 26 on "Billboard"'s Hot 100 in 1991. Subsequent singles from "Anything Is Possible" failed to chart on the Hot 100, although "One Step Ahead" scored on the Hot Maxi Singles and Hot Dance charts, peaking at numbers 21 and 18, respectively.


title: Debbie Gibson

text: During this time, Gibson was part of the supergroup that recorded the charity single "Voices That Care", which peaked at number 11 on the Hot 100 chart.


title: Debbie Gibson

text: In 1992, she contributed a version of "Sleigh Ride" to the charity album "A Very Special Christmas 2" which hit #7 on the Billboard album chart and was certified Double Platinum for shipment of two million copies.


title: Debbie Gibson

text: In 1995, she signed with EMI's SBK Records division and recorded her only album for the label, "Think with Your Heart." It was an adult contemporary-heavy album consisting of piano and keyboard ballads recorded predominantly with the London Philharmonic Orchestra. The album's producer, Niko Bolas, who was usually Neil Young's co-producer, was producing the reunion album for Circle Jerks (a veteran punk band) and invited Gibson to a recording session for that band's album. She sang background vocals on the song "I Wanna Destroy You", as well as appearing at and participating in the Circle Jerks' performance at the punk venue CBGB wearing one of the band's T-shirts and sharing a microphone with frontman Keith Morris.


title: Debbie Gibson

text: In 1998, she sang the song, "I Do", which is featured on the soundtrack to the film, "The Naked Man". The lyrics to the song were composed by the co-writer of the film, Ethan Coen. The soundtrack has never been released.


title: Debbie Gibson

text: After parting company with EMI, Gibson formed her own record label, Espiritu, to release her original material. Her sixth album, "Deborah" (1997), marked her full return to pop. "Deborah" includes the lead single "Only Words". "Only Words" (Dance Edit) became a Top-40 Hot Dance Music/Club Play hit. The album's other single was the ballad "Naturally". Though it only sold 20,000 in the US, "Deborah" remains well-respected.


title: Debbie Gibson

text: In 2001, she released her seventh album on her new record label, Golden Egg, titled "M.Y.O.B." It featured three singles: the sensual pop song "What You Want", the Latin-infused dance-pop song "Your Secret", and the bass-heavy "M.Y.O.B." Highlights from the album include the sultry Latin-flavored smooth jazz song "In Blue", a vintage-style ballad "Wishing You Were Here", "Jaded", and a remix of "M.Y.O.B." with the background vocals of her two nieces.


title: Debbie Gibson

text: In 2005, Gibson co-wrote and recorded a song titled "Someone You Love" with the O'Neill Brothers. With the brothers, she released an updated, acoustic version of her number-one hit "Lost in Your Eyes". An Emmy-nominated PBS special aired in 2005.


title: Debbie Gibson

text: The March 2005 issue of "Playboy" featured a nude pictorial of Gibson, coinciding with the release of her single, "Naked". She has said that the magazine had asked her five times to pose for them since she turned 18. She agreed to pose to revamp her image, describing how one casting call called her agent, not realizing that Gibson had long since outgrown her teenager image. The single peaked at No. 35 on the "Billboard" Hot Single Sales chart in March 2005.


title: Debbie Gibson

text: In 2006, Gibson went on tour with the O'Neill Brothers for "Someone You Love Tour" in 2006. "Lost in Your Eyes" was revamped with an acoustic sound and a song "Someone You Love" was written and performed by Gibson and the O'Neill Brothers. These two songs also appeared on the O'Neill Brothers album "Someone You Love".


title: Debbie Gibson

text: She had a resurgence of popularity in niche markets. Her single "Your Secret" came back from its dormant state and became popular on some radio stations, including Super 91.7 WMPH in Wilmington, Delaware. "Your Secret" has been on their request show, "Total Control Radio", for 12 months; it reached No. 1 on its third week on that station's chart in May 2006. It charted along with a few of her other singles, "M.Y.O.B." and "Only Words" (Dance Edit), the Eurodance mix. They all have become recurrent hits on WMPH.


title: Debbie Gibson

text: The 2006 single, "Say Goodbye", featuring dance-pop artist Jordan Knight, made an impression in the Soft AC and Hot AC radio formats, becoming the third-most added single during summer, 2006. It debuted at number 35 on the Hot Contemporary chart, peaking at number 24 in early September. The same year, Gibson appeared in the independent film "Coffee Date" with Wilson Cruz and Jonathan Silverman and provided a soundtrack song called "Sounds Like Love".


title: Debbie Gibson

text: On November 14, 2006, Gibson released the song "Famous" on her official website. The song was written by Gibson and Tiziano Lugli and was produced by Lugli.


title: Debbie Gibson

text: In May 2007, the world premiere of "Electric Youth: The Musical" was unveiled at the Starlight Theatre in Orlando, Florida. The musical featured 14 of Gibson's songs and was directed by Dean Parker. On August 24, 2007, Gibson and Frankie Avalon hosted "Time-Life Presents Dick Clark's American Bandstand 50th Anniversary Collection".


title: Debbie Gibson

text: In September 2007, Gibson considered creating a camp on the West Coast. She is the founder and creator of Camp Electric Youth, a children's summer day camp, which ran from July 7–18, 2008. It claims to be the first camp of its kind in the Los Angeles area. The camp was reportedly attended by "over 120 talented singers, actors, and dancers" from around the world.


title: Debbie Gibson

text: Gibson was a judge for the online talent competition, Total Pop Star, along with Andrew Van Slee (producer and judge) and Joey Lawrence (from "Blossom"). The first season ran from November 12, 2007 – May 30, 2008, though it was later extended to June. The show ended abruptly during its second season.


title: Debbie Gibson

text: In May 2008, Gibson performed her 1980s hits, along with her Broadway role songs, during a three-run week at Harrah's in Atlantic City, New Jersey.


title: Debbie Gibson

text: She later appeared on the April 2008 cover of "Lavender Magazine" (an LGBT magazine in Minnesota) and was interviewed about her career and upcoming projects. Then on the 24th, Gibson hosted and performed on Spotlightlive '80s Karaoke Experience in New York singing songs such as "Only in My Dreams", "Out of the Blue", "Love Shack" (an original hit for the B-52's) and "9 to 5". She performed with Samantha Fox, Tiffany, & Rick Astley at the Colisée Pepsi in Quebec City, Canada, on April 10, 2009.


title: Debbie Gibson

text: On March 9, 2009, Gibson released a new song called "Already Gone" on her official website and reverberation, written by Gibson and produced by Fred Coury. It was accompanied by a music video, released on March 13, 2009, produced by Guy Birtwhistle and directed by John Knowles, which starred Birtwhistle, Gibson, and Steve Valentine.


title: Debbie Gibson

text: In January 2010, an unofficial clip named "Cougar" was uploaded on YouTube. She also became a spokesperson for Murad's Resurgence Skin Care and plays piano and sings a line of the song called "Cougar".


title: Debbie Gibson

text: Gibson's song "Rise", from the documentary "3 Billion and Counting", was included on the shortlist for an Academy Award for Best Song in a Film in 2010. In the summer of 2011, she released "Rise" on iTunes, and also performed it on "Good Morning America" in New York.


title: Debbie Gibson

text: Gibson performed as Mother Nature in Cirque Dreams Jungle Fantasy at Foxwoods/MGM Casino from July 27 to September 1, 2010.


title: Debbie Gibson

text: In 2010, the album "Ms. Vocalist", released via Sony Japan, reached the Top 10 on the Japanese "Billboard" chart. Gibson covered J-Pop tunes for the album that were originally sung by Japanese artists such as Chage and Aska ("Say Yes"), Yutaka Ozaki ("I love you"), Sekaiju no Dareyori Kitto (by Miho Nakayama and WANDS) among others, plus Japanese/English version of her number-one hit "Lost in Your Eyes" and a duet with Eric Martin. The first single from the album, "I Love You", hit No. 1 on the international cable radio chart. On November 3, 2010, "I Love You", the first single from her 2010 album , reached number one


title: Debbie Gibson

text: In January 2011, Gibson wrote, performed, and produced the song "Snake Charmer" for the film "Mega Python vs. Gatoroid".


title: Debbie Gibson

text: During the summer of 2011 Gibson toured with fellow 1980s pop princess Tiffany Darwish. In June 2011, Gibson appeared in Katy Perry's music video "Last Friday Night (T.G.I.F.)" alongside several other guest stars.


title: Debbie Gibson

text: On August 27, 2016, Gibson starred in an original Hallmark Channel film, "Summer of Dreams", about a former pop star, trying to make a comeback, who finds herself better suited as a school's choir director. She also recorded a song titled "Wonderland" for the film.


title: Debbie Gibson

text: In June 2017, "Billboard" Gibson achieved her highest-charting hit in more than 25 years in her duet with Sir Ivan on "I Am Peaceman", which hit number 26 on the "Billboard "Dance Club chart.


title: Debbie Gibson

text: In June 2018, Gibson appeared in the music video of American heavy-metal band Voices of Extreme's cover version of "Foolish Beat".


title: Debbie Gibson

text: In March 2019, Gibson hosted a special program on SiriusXM Radio's '80s on 8 channel to celebrate the 30th anniversary of her "Electric Youth" album, during which she played each song from the album in sequence, accompanied by personal stories surrounding each song.


title: Debbie Gibson

text: On May 2, the Mixtape Tour commenced in Cincinnati, Ohio. Performers on this tour include Gibson, Tiffany, Salt-N-Pepa, and Naughty by Nature, with New Kids on the Block billed as the headline performers. The tour grossed $53.2 million and sold 662,911 tickets over 55 dates.


title: Debbie Gibson

text: On June 7, 2019, Gibson released a new pop anthem "Girls Night Out". The music video for "Girls Night Out" was shot in Las Vegas, Nevada. The song peaked at number four on the" Billboard" Dance Club chart.


title: Debbie Gibson

text: On August 20, 2021, Gibson released the album "The Body Remembers", her first studio recording of original songs since "M.Y.O.B.". It contained a new version of "Lost in Your Eyes" with Joey McIntyre.


title: Debbie Gibson

text: Gibson debuted on Broadway in 1992, playing Eponine in "Les Misérables". She then went to London and starred as Sandy in "Grease"—a role for which 800 other people tried out before producers chose Gibson—in a West End production. The show broke box office sales records. The single version of "You're the One That I Want", a duet with Craig McLachlan, taken from the original cast recording, reached number 13 on the UK Singles Chart in 1993.


title: Debbie Gibson

text: On returning to the States, she appeared in the Broadway touring production, this time playing Rizzo. She played Fanny Brice in a revamped "Funny Girl" tour. She has had many successful theatre credits; she was among the many actresses who took the starring role of Belle in the Broadway production of "Beauty and the Beast". She replaced Kerry Butler in September 1997 and was in the show until June 1998, when Kim Huber then succeeded her. She also starred in the critically lauded production of "" (in a production staged at the Paper Mill Playhouse). Gibson starred as Louise opposite Broadway legend Betty Buckley. She participated in the national tour of "Joseph and the Amazing Technicolor Dreamcoat", where she played the Narrator, and starred as Cinderella in the national tour of Rodgers & Hammerstein's musical with Eartha Kitt as the Fairy Godmother. In October 2002, she starred as Velma Kelly in the Boston production of "Chicago". In 2003, she played Sally Bowles in the Broadway revival of "Cabaret". From March to April 2004, she played the role of Marta in the UCLA Reprise! production of "Company".


title: Debbie Gibson

text: Gibson starred as Anna Leonowens in Cabrillo Music Theatre's production of the Richard Rodgers and Oscar Hammerstein II musical "The King and I", which began October 17, 2008, in the Kavli Theatre at the Thousand Oaks Civic Arts Plaza and ran through October 26.


title: Debbie Gibson

text: Gibson co-hosted Nickelodeon's first Kids' Choice Awards in 1988, alongside Tony Danza, Brian Robbins, and Dan Schneider.


title: Debbie Gibson

text: Gibson co-starred with actor Lorenzo Lamas in the low-budget action/adventure film "Mega Shark Versus Giant Octopus", produced by The Asylum and released on May 19, 2009. The film drew in 2 million viewers on Syfy in 2009. Its trailer became a viral hit, scoring over one million hits on MTV.com and YouTube. The film premiered at the 2009 Cannes Film Festival. Gibson's former music rival Tiffany had her film "Necrosis" (or "Blood Snow") premiere at the Cannes, as well. Gibson and Tiffany starred in a Syfy original movie entitled "Mega Python vs. Gatoroid", aired on January 29, 2011. The pairing was suggested by Tiffany, who wanted to play-off their supposed rivalry. Gibson reprised her role as Emma McNeil in the 2014 film "Mega Shark Versus Mecha Shark".


title: Debbie Gibson

text: She starred in the UP TV movie called "The Music in Me" alongside Gloria Reuben in 2015. The film also featured an original song called "Promises", written and performed by Gibson.


title: Debbie Gibson

text: In mid-2003, Gibson was a judge on the "American Idol" spin-off "American Juniors", which lasted one season. In January 2006, she joined the cast of "Skating with Celebrities" on Fox Television, partnered with Canadian former world-champion figure skater Kurt Browning. She was voted out in the third episode.


title: Debbie Gibson

text: Gibson competed on the fifth season of "The Celebrity Apprentice", which began airing on February 19, 2012. On the fourth task, she won $50,000 for her charity, Children International. Gibson was fired on April 1, 2012, in the seventh task because she had brought in the least amount of money between her teammates in the boardroom, Dayana Mendoza and Teresa Giudice. While both Mendoza and Giudice were arguably weaker candidates going forward, given that Mendoza was cited as the women's weakest link and Giudice had failed the task as project manager, Gibson was fired, instead.


title: Debbie Gibson

text: In 2017, Gibson was one of the celebrities competing on the 25th season of "Dancing with the Stars". She was paired with first-time pro-dancer, Alan Bersten. On September 26, 2017, Gibson and Bersten were the second couple eliminated coming in 12th place.


title: Debbie Gibson

text: In September 2018, Gibson starred in the Hallmark Channel film "Wedding of Dreams," a sequel to 2016's "Summer of Dreams."


title: Debbie Gibson

text: Gibson made a guest appearance on season 5 episode 10 of the Netflix series "Lucifer", which premiered on May 28, 2021. In the episode titled "Bloody Celestial Karaoke Jam", she plays a mother who sings "Every Breath You Take" with Lucifer (Tom Ellis) in an interrogation room.


title: Debbie Gibson

text: In tandem with the second album, she created a perfume called Electric Youth that was distributed by Revlon, as well as other makeup essentials for young girls that were distributed nationwide through Natural Wonder Cosmetics. Gibson's trademark was her hats, usually a black pork-pie style. She also made popular wearing tight, rolled-up jeans, vests over a T-shirt, friendship bracelets and two Swatch watches as on the back cover of her popular album "Electric Youth" and in her music video "Staying Together". Her influences were Madonna and Olivia Newton-John, though she has often stated she admires Elton John and Billy Joel as favorite artists and was asked to sing and perform live with both at the former's Madison Square Garden concert, which she did. Gibson appeared on the covers of numerous teen magazines such as "Tiger Beat".


title: Debbie Gibson

text: Over the years, Gibson has been the target of stalkers. Robert Bardo, who was convicted of murdering actress Rebecca Schaeffer in 1989, had a wall in his house adorned with pictures of Gibson and Tiffany Darwish. In May 1998, Michael Falkner, a disgruntled fan from Eau Claire, Wisconsin, was arrested outside Manhattan's Palace Theater, where Gibson was performing in the live-musical adaptation of Disney's "Beauty and the Beast". This was after Gibson received threatening letters, emails, and faxes from Falkner, who used the alias 'Starcade'. In 2008, Gibson filed for a restraining order against Spanish taxi driver Jorge Puigdollers, who had stalked her since 2002. However, a temporary restraining order was not issued, and a court date was set to determine if a restraining order was appropriate. The proceeding was dismissed when Gibson failed to show up for the hearing.


title: Debbie Gibson

text: Gibson was once engaged to Jonathan Kanterman and was in a relationship with Rutledge Taylor from 2008 to 2019.


title: Debbie Gibson

text: In 2014, in response to fans' concern about her weight loss, Gibson stated she had developed symptoms of Lyme disease in early 2013.


title: Debbie Gibson

text: In 2016, Gibson appeared on "", where she spoke out about her drug use in the past, after the death of singer Prince:


title: Debbie Gibson

text: I really feel like I haven't fully articulated it till now and really spoke candidly till now... When I heard the news about Prince and the fact that it might have been prescription drug-related, I really had a moment of, like, "That's awful and that's sad – and I can relate." And, unfortunately, 90% of the entertainment community can relate. I remember being on the road at, like, 25, touring with theater and doing my own cocktail of Tylenol PM and Xanax. It's like, "Oh, I found a way to make the Xanax last longer with the Tylenol PM. I mean, it's as simple as that, and that is how performers get in so much trouble."


title: Beta distribution

text: In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] in terms of two positive parameters, denoted by "alpha" ("α") and "beta" ("β"), that appear as exponents of the random variable and control the shape of the distribution.


title: Beta distribution

text: The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions.


title: Beta distribution

text: In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions.


title: Beta distribution

text: The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas "beta distribution of the second kind" is an alternative name for the beta prime distribution. The generalization to multiple variables is called a Dirichlet distribution.


title: Beta distribution

text: The probability density function (PDF) of the beta distribution, for , and shape parameters "α", "β" > 0, is a power function of the variable "x" and of its reflection as follows:


title: Beta distribution

text: where Γ("z") is the gamma function. The beta function, formula_22, is a normalization constant to ensure that the total probability is 1. In the above equations "x" is a realization—an observed value that actually occurred—of a random process "X".


title: Beta distribution

text: This definition includes both ends and , which is consistent with definitions for other continuous distributions supported on a bounded interval which are special cases of the beta distribution, for example the arcsine distribution, and consistent with several authors, like N. L. Johnson and S. Kotz. However, the inclusion of and does not work for ; accordingly, several other authors, including W. Feller, choose to exclude the ends and , (so that the two ends are not actually part of the domain of the density function) and consider instead .


title: Beta distribution

text: Several authors, including N. L. Johnson and S. Kotz, use the symbols "p" and "q" (instead of "α" and "β") for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters "α" and "β" approach the value of zero.


title: Beta distribution

text: In the following, a random variable "X" beta-distributed with parameters "α" and "β" will be denoted by:


title: Beta distribution

text: Other notations for beta-distributed random variables used in the statistical literature are formula_24 and formula_25.


title: Beta distribution

text: where formula_27 is the incomplete beta function and formula_28 is the regularized incomplete beta function.


title: Beta distribution

text: The beta distribution may also be reparameterized in terms of its mean "μ" and the sum of the two shape parameters ( p. 83). Denoting by αPosterior and βPosterior the shape parameters of the posterior beta distribution resulting from applying Bayes theorem to a binomial likelihood function and a prior probability, the interpretation of the addition of both shape parameters to be sample size = "ν" = "α"·Posterior + "β"·Posterior is only correct for the Haldane prior probability Beta(0,0). Specifically, for the Bayes (uniform) prior Beta(1,1) the correct interpretation would be sample size = "α"·Posterior + "β" Posterior − 2, or "ν" = (sample size) + 2. For sample size much larger than 2, the difference between these two priors becomes negligible. (See section Bayesian inference for further details.) ν = α + β is referred to as the "sample size" of a Beta distribution, but one should remember that it is, strictly speaking, the "sample size" of a binomial likelihood function only when using a Haldane Beta(0,0) prior in Bayes theorem.


title: Beta distribution

text: This parametrization may be useful in Bayesian parameter estimation. For example, one may administer a test to a number of individuals. If it is assumed that each person's score (0 ≤ "θ" ≤ 1) is drawn from a population-level Beta distribution, then an important statistic is the mean of this population-level distribution. The mean and sample size parameters are related to the shape parameters α and β via


title: Beta distribution

text: Under this parametrization, one may place an uninformative prior probability over the mean, and a vague prior probability (such as an exponential or gamma distribution) over the positive reals for the sample size, if they are independent, and prior data and/or beliefs justify it.


title: Beta distribution

text: Concave beta distributions, which have formula_29, can be parametrized in terms of mode and "concentration". The mode, formula_30, and concentration, formula_31, can be used to define the usual shape parameters as follows:


title: Beta distribution

text: For the mode, formula_33, to be well-defined, we need formula_29, or equivalently formula_35. If instead we define the concentration as formula_36, the condition simplifies to formula_37 and the beta density at formula_38 and formula_39 can be written as:


title: Beta distribution

text: where formula_41 directly scales the sufficient statistics, formula_42 and formula_43. Note also that in the limit, formula_44, the distribution becomes flat.


title: Beta distribution

text: Solving the system of (coupled) equations given in the above sections as the equations for the mean and the variance of the beta distribution in terms of the original parameters "α" and "β", one can express the "α" and "β" parameters in terms of the mean ("μ") and the variance (var):


title: Beta distribution

text: This parametrization of the beta distribution may lead to a more intuitive understanding than the one based on the original parameters "α" and "β". For example, by expressing the mode, skewness, excess kurtosis and differential entropy in terms of the mean and the variance:


title: Beta distribution

text: A beta distribution with the two shape parameters α and β is supported on the range [0,1] or (0,1). It is possible to alter the location and scale of the distribution by introducing two further parameters representing the minimum, "a", and maximum "c" ("c" > "a"), values of the distribution, by a linear transformation substituting the non-dimensional variable "x" in terms of the new variable "y" (with support ["a","c"] or ("a","c")) and the parameters "a" and "c":


title: Beta distribution

text: The probability density function of the four parameter beta distribution is equal to the two parameter distribution, scaled by the range ("c"-"a"), (so that the total area under the density curve equals a probability of one), and with the "y" variable shifted and scaled as follows:


title: Beta distribution

text: That a random variable "Y" is Beta-distributed with four parameters α, β, "a", and "c" will be denoted by:


title: Beta distribution

text: Note: the geometric mean and harmonic mean cannot be transformed by a linear transformation in the way that the mean, median and mode can.


title: Beta distribution

text: The statistical dispersion measures are scaled (they do not need to be shifted because they are already centered on the mean) by the range (c-a), linearly for the mean deviation and nonlinearly for the variance:


title: Beta distribution

text: Since the skewness and excess kurtosis are non-dimensional quantities (as moments centered on the mean and normalized by the standard deviation), they are independent of the parameters "a" and "c", and therefore equal to the expressions given above in terms of "X" (with support [0,1] or (0,1)):


title: Beta distribution

text: The mode of a Beta distributed random variable "X" with "α", "β" > 1 is the most likely value of the distribution (corresponding to the peak in the PDF), and is given by the following expression:


title: Beta distribution

text: When both parameters are less than one ("α", "β" < 1), this is the anti-mode: the lowest point of the probability density curve.


title: Beta distribution

text: Letting "α" = "β", the expression for the mode simplifies to 1/2, showing that for "α" = "β" > 1 the mode (resp. anti-mode when ), is at the center of the distribution: it is symmetric in those cases. See Shapes section in this article for a full list of mode cases, for arbitrary values of "α" and "β". For several of these cases, the maximum value of the density function occurs at one or both ends. In some cases the (maximum) value of the density function occurring at the end is finite. For example, in the case of "α" = 2, "β" = 1 (or "α" = 1, "β" = 2), the density function becomes a right-triangle distribution which is finite at both ends. In several other cases there is a singularity at one end, where the value of the density function approaches infinity. For example, in the case "α" = "β" = 1/2, the Beta distribution simplifies to become the arcsine distribution. There is debate among mathematicians about some of these cases and whether the ends ("x" = 0, and "x" = 1) can be called "modes" or not.


title: Beta distribution

text: The median of the beta distribution is the unique real number formula_57 for which the regularized incomplete beta function formula_58. There is no general closed-form expression for the median of the beta distribution for arbitrary values of "α" and "β". Closed-form expressions for particular values of the parameters "α" and "β" follow:


title: Beta distribution

text: The following are the limits with one parameter finite (non-zero) and the other approaching these limits:


title: Beta distribution

text: A reasonable approximation of the value of the median of the beta distribution, for both α and β greater or equal to one, is given by the formula


title: Beta distribution

text: When α, β ≥ 1, the relative error (the absolute error divided by the median) in this approximation is less than 4% and for both α ≥ 2 and β ≥ 2 it is less than 1%. The absolute error divided by the difference between the mean and the mode is similarly small:


title: Beta distribution

text: The expected value (mean) ("μ") of a Beta distribution random variable "X" with two parameters "α" and "β" is a function of only the ratio "β"/"α" of these parameters:


title: Beta distribution

text: Letting in the above expression one obtains , showing that for the mean is at the center of the distribution: it is symmetric. Also, the following limits can be obtained from the above expression:


title: Beta distribution

text: Therefore, for "β"/"α" → 0, or for "α"/"β" → ∞, the mean is located at the right end, . For these limit ratios, the beta distribution becomes a one-point degenerate distribution with a Dirac delta function spike at the right end, , with probability 1, and zero probability everywhere else. There is 100% probability (absolute certainty) concentrated at the right end, .


title: Beta distribution

text: Similarly, for "β"/"α" → ∞, or for "α"/"β" → 0, the mean is located at the left end, . The beta distribution becomes a 1-point Degenerate distribution with a Dirac delta function spike at the left end, "x" = 0, with probability 1, and zero probability everywhere else. There is 100% probability (absolute certainty) concentrated at the left end, "x" = 0. Following are the limits with one parameter finite (non-zero) and the other approaching these limits:


title: Beta distribution

text: While for typical unimodal distributions (with centrally located modes, inflexion points at both sides of the mode, and longer tails) (with Beta("α", "β") such that ) it is known that the sample mean (as an estimate of location) is not as robust as the sample median, the opposite is the case for uniform or "U-shaped" bimodal distributions (with Beta("α", "β") such that ), with the modes located at the ends of the distribution. As Mosteller and Tukey remark ( p. 207) "the average of the two extreme observations uses all the sample information. This illustrates how, for short-tailed distributions, the extreme observations should get more weight." By contrast, it follows that the median of "U-shaped" bimodal distributions with modes at the edge of the distribution (with Beta("α", "β") such that ) is not robust, as the sample median drops the extreme sample observations from consideration. A practical application of this occurs for example for random walks, since the probability for the time of the last visit to the origin in a random walk is distributed as the arcsine distribution Beta(1/2, 1/2): the mean of a number of realizations of a random walk is a much more robust estimator than the median (which is an inappropriate sample measure estimate in this case).


title: Beta distribution

text: The logarithm of the geometric mean "G" of a distribution with random variable "X" is the arithmetic mean of ln("X"), or, equivalently, its expected value:


title: Beta distribution

text: Therefore, the geometric mean of a beta distribution with shape parameters "α" and "β" is the exponential of the digamma functions of "α" and "β" as follows:


title: Beta distribution

text: While for a beta distribution with equal shape parameters α = β, it follows that skewness = 0 and mode = mean = median = 1/2, the geometric mean is less than 1/2: . The reason for this is that the logarithmic transformation strongly weights the values of "X" close to zero, as ln("X") strongly tends towards negative infinity as "X" approaches zero, while ln("X") flattens towards zero as .


title: Beta distribution

text: Following are the limits with one parameter finite (non-zero) and the other approaching these limits:


title: Beta distribution

text: The accompanying plot shows the difference between the mean and the geometric mean for shape parameters α and β from zero to 2. Besides the fact that the difference between them approaches zero as α and β approach infinity and that the difference becomes large for values of α and β approaching zero, one can observe an evident asymmetry of the geometric mean with respect to the shape parameters α and β. The difference between the geometric mean and the mean is larger for small values of α in relation to β than when exchanging the magnitudes of β and α.


title: Beta distribution

text: N. L.Johnson and S. Kotz suggest the logarithmic approximation to the digamma function "ψ"("α") ≈ ln("α" − 1/2) which results in the following approximation to the geometric mean:


title: Beta distribution

text: Numerical values for the relative error in this approximation follow: []; []; []; []; []; []; []; [].


title: Beta distribution

text: Similarly, one can calculate the value of shape parameters required for the geometric mean to equal 1/2. Given the value of the parameter "β", what would be the value of the other parameter, "α", required for the geometric mean to equal 1/2?. The answer is that (for ), the value of "α" required tends towards as . For example, all these couples have the same geometric mean of 1/2: [], [], [], [], [], [], [].


title: Beta distribution

text: The fundamental property of the geometric mean, which can be proven to be false for any other mean, is


title: Beta distribution

text: This makes the geometric mean the only correct mean when averaging "normalized" results, that is results that are presented as ratios to reference values. This is relevant because the beta distribution is a suitable model for the random behavior of percentages and it is particularly suitable to the statistical modelling of proportions. The geometric mean plays a central role in maximum likelihood estimation, see section "Parameter estimation, maximum likelihood." Actually, when performing maximum likelihood estimation, besides the geometric mean "G" based on the random variable X, also another geometric mean appears naturally: the geometric mean based on the linear transformation ––, the mirror-image of "X", denoted by "G":


title: Beta distribution

text: Although both "G" and "G" are asymmetric, in the case that both shape parameters are equal , the geometric means are equal: "G" = "G". This equality follows from the following symmetry displayed between both geometric means:


title: Beta distribution

text: The inverse of the harmonic mean ("H") of a distribution with random variable "X" is the arithmetic mean of 1/"X", or, equivalently, its expected value. Therefore, the harmonic mean ("H") of a beta distribution with shape parameters "α" and "β" is:


title: Beta distribution

text: The harmonic mean ("H") of a Beta distribution with "α" < 1 is undefined, because its defining expression is not bounded in [0, 1] for shape parameter "α" less than unity.


title: Beta distribution

text: showing that for "α" = "β" the harmonic mean ranges from 0, for "α" = "β" = 1, to 1/2, for "α" = "β" → ∞.


title: Beta distribution

text: The harmonic mean plays a role in maximum likelihood estimation for the four parameter case, in addition to the geometric mean. Actually, when performing maximum likelihood estimation for the four parameter case, besides the harmonic mean "H" based on the random variable "X", also another harmonic mean appears naturally: the harmonic mean based on the linear transformation (1 − "X"), the mirror-image of "X", denoted by "H":


title: Beta distribution

text: The harmonic mean ("H") of a Beta distribution with "β" < 1 is undefined, because its defining expression is not bounded in [0, 1] for shape parameter "β" less than unity.


title: Beta distribution

text: Although both "H" and "H" are asymmetric, in the case that both shape parameters are equal "α" = "β", the harmonic means are equal: "H" = "H". This equality follows from the following symmetry displayed between both harmonic means:


title: Beta distribution

text: The variance (the second moment centered on the mean) of a Beta distribution random variable "X" with parameters α and β is:


title: Beta distribution

text: showing that for "α" = "β" the variance decreases monotonically as increases. Setting in this expression, one finds the maximum variance var("X") = 1/4 which only occurs approaching the limit, at .


title: Beta distribution

text: The beta distribution may also be parametrized in terms of its mean "μ" and sample size () (see subsection Mean and sample size):


title: Beta distribution

text: Using this parametrization, one can express the variance in terms of the mean "μ" and the sample size "ν" as follows:


title: Beta distribution

text: Also, the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:


title: Beta distribution

text: The logarithm of the geometric variance, ln(var), of a distribution with random variable "X" is the second moment of the logarithm of "X" centered on the geometric mean of "X", ln("G"):


title: Beta distribution

text: In the Fisher information matrix, and the curvature of the log likelihood function, the logarithm of the geometric variance of the reflected variable 1 − "X" and the logarithm of the geometric covariance between "X" and 1 − "X" appear:


title: Beta distribution

text: For a beta distribution, higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two Gamma distributions and differentiating through the integral. They can be expressed in terms of higher order poly-gamma functions. See the section . The variance of the logarithmic variables and covariance of ln "X" and ln(1−"X") are:


title: Beta distribution

text: where the trigamma function, denoted ψ(α), is the second of the polygamma functions, and is defined as the derivative of the digamma function:


title: Beta distribution

text: The accompanying plots show the log geometric variances and log geometric covariance versus the shape parameters "α" and "β". The plots show that the log geometric variances and log geometric covariance are close to zero for shape parameters α and β greater than 2, and that the log geometric variances rapidly rise in value for shape parameter values "α" and "β" less than unity. The log geometric variances are positive for all values of the shape parameters. The log geometric covariance is negative for all values of the shape parameters, and it reaches large negative values for "α" and "β" less than unity.


title: Beta distribution

text: Following are the limits with one parameter finite (non-zero) and the other approaching these limits:


title: Beta distribution

text: Although both ln(var) and ln(var) are asymmetric, when the shape parameters are equal, α = β, one has: ln(var) = ln(var). This equality follows from the following symmetry displayed between both log geometric variances:


title: Beta distribution

text: The mean absolute deviation around the mean for the beta distribution with shape parameters α and β is:


title: Beta distribution

text: The mean absolute deviation around the mean is a more robust estimator of statistical dispersion than the standard deviation for beta distributions with tails and inflection points at each side of the mode, Beta("α", "β") distributions with "α","β" > 2, as it depends on the linear (absolute) deviations rather than the square deviations from the mean. Therefore, the effect of very large deviations from the mean are not as overly weighted.


title: Beta distribution

text: Using Stirling's approximation to the Gamma function, N.L.Johnson and S.Kotz derived the following approximation for values of the shape parameters greater than unity (the relative error for this approximation is only −3.5% for "α" = "β" = 1, and it decreases to zero as "α" → ∞, "β" → ∞):


title: Beta distribution

text: At the limit α → ∞, β → ∞, the ratio of the mean absolute deviation to the standard deviation (for the beta distribution) becomes equal to the ratio of the same measures for the normal distribution: formula_107. For α = β = 1 this ratio equals formula_108, so that from α = β = 1 to α, β → ∞ the ratio decreases by 8.5%. For α = β = 0 the standard deviation is exactly equal to the mean absolute deviation around the mean. Therefore, this ratio decreases by 15% from α = β = 0 to α = β = 1, and by 25% from α = β = 0 to α, β → ∞ . However, for skewed beta distributions such that α → 0 or β → 0, the ratio of the standard deviation to the mean absolute deviation approaches infinity (although each of them, individually, approaches zero) because the mean absolute deviation approaches zero faster than the standard deviation.


title: Beta distribution

text: one can express the mean absolute deviation around the mean in terms of the mean μ and the sample size ν as follows:


title: Beta distribution

text: The skewness (the third moment centered on the mean, normalized by the 3/2 power of the variance) of the beta distribution is


title: Beta distribution

text: Letting α = β in the above expression one obtains γ = 0, showing once again that for α = β the distribution is symmetric and hence the skewness is zero. Positive skew (right-tailed) for α < β, negative skew (left-tailed) for α > β.


title: Beta distribution

text: The accompanying plot of skewness as a function of variance and mean shows that maximum variance (1/4) is coupled with zero skewness and the symmetry condition (μ = 1/2), and that maximum skewness (positive or negative infinity) occurs when the mean is located at one end or the other, so that the "mass" of the probability distribution is concentrated at the ends (minimum variance).


title: Beta distribution

text: The following expression for the square of the skewness, in terms of the sample size ν = α + β and the variance "var", is useful for the method of moments estimation of four parameters:


title: Beta distribution

text: This expression correctly gives a skewness of zero for α = β, since in that case (see ): formula_119.


title: Beta distribution

text: For the asymmetric cases (α ≠ β) the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:


title: Beta distribution

text: The beta distribution has been applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported to be a good indicator of the condition of a gear. Kurtosis has also been used to distinguish the seismic signal generated by a person's footsteps from other signals. As persons or other targets moving on the ground generate continuous signals in the form of seismic waves, one can separate different targets based on the seismic waves they generate. Kurtosis is sensitive to impulsive signals, so it's much more sensitive to the signal generated by human footsteps than other signals generated by vehicles, winds, noise, etc. Unfortunately, the notation for kurtosis has not been standardized. Kenney and Keeping use the symbol γ for the excess kurtosis, but Abramowitz and Stegun use different terminology. To prevent confusion between kurtosis (the fourth moment centered on the mean, normalized by the square of the variance) and excess kurtosis, when using symbols, they will be spelled out as follows:


title: Beta distribution

text: Therefore, for symmetric beta distributions, the excess kurtosis is negative, increasing from a minimum value of −2 at the limit as {α = β} → 0, and approaching a maximum value of zero as {α = β} → ∞. The value of −2 is the minimum value of excess kurtosis that any distribution (not just beta distributions, but any distribution of any possible kind) can ever achieve. This minimum value is reached when all the probability density is entirely concentrated at each end "x" = 0 and "x" = 1, with nothing in between: a 2-point Bernoulli distribution with equal probability 1/2 at each end (a coin toss: see section below "Kurtosis bounded by the square of the skewness" for further discussion). The description of kurtosis as a measure of the "potential outliers" (or "potential rare, extreme values") of the probability distribution, is correct for all distributions including the beta distribution. When rare, extreme values can occur in the beta distribution, the higher its kurtosis; otherwise, the kurtosis is lower. For α ≠ β, skewed beta distributions, the excess kurtosis can reach unlimited positive values (particularly for α → 0 for finite β, or for β → 0 for finite α) because the side away from the mode will produce occasional extreme values. Minimum kurtosis takes place when the mass density is concentrated equally at each end (and therefore the mean is at the center), and there is no probability mass density in between the ends.


title: Beta distribution

text: The excess kurtosis can also be expressed in terms of just the following two parameters: the variance "var", and the sample size ν as follows:


title: Beta distribution

text: The plot of excess kurtosis as a function of the variance and the mean shows that the minimum value of the excess kurtosis (−2, which is the minimum possible value for excess kurtosis for any distribution) is intimately coupled with the maximum value of variance (1/4) and the symmetry condition: the mean occurring at the midpoint (μ = 1/2). This occurs for the symmetric case of α = β = 0, with zero skewness. At the limit, this is the 2 point Bernoulli distribution with equal probability 1/2 at each Dirac delta function end "x" = 0 and "x" = 1 and zero probability everywhere else. (A coin toss: one face of the coin being "x" = 0 and the other face being "x" = 1.) Variance is maximum because the distribution is bimodal with nothing in between the two modes (spikes) at each end. Excess kurtosis is minimum: the probability density "mass" is zero at the mean and it is concentrated at the two peaks at each end. Excess kurtosis reaches the minimum possible value (for any distribution) when the probability density function has two spikes at each end: it is bi-"peaky" with nothing in between them.


title: Beta distribution

text: On the other hand, the plot shows that for extreme skewed cases, where the mean is located near one or the other end (μ = 0 or μ = 1), the variance is close to zero, and the excess kurtosis rapidly approaches infinity when the mean of the distribution approaches either end.


title: Beta distribution

text: Alternatively, the excess kurtosis can also be expressed in terms of just the following two parameters: the square of the skewness, and the sample size ν as follows:


title: Beta distribution

text: From this last expression, one can obtain the same limits published practically a century ago by Karl Pearson in his paper, for the beta distribution (see section below titled "Kurtosis bounded by the square of the skewness"). Setting α + β= ν = 0 in the above expression, one obtains Pearson's lower boundary (values for the skewness and excess kurtosis below the boundary (excess kurtosis + 2 − skewness = 0) cannot occur for any distribution, and hence Karl Pearson appropriately called the region below this boundary the "impossible region"). The limit of α + β = ν → ∞ determines Pearson's upper boundary.


title: Beta distribution

text: Values of ν = α + β such that ν ranges from zero to infinity, 0 < ν < ∞, span the whole region of the beta distribution in the plane of excess kurtosis versus squared skewness.


title: Beta distribution

text: For the unsymmetric cases (α ≠ β) the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:


title: Beta distribution

text: The characteristic function is the Fourier transform of the probability density function. The characteristic function of the beta distribution is Kummer's confluent hypergeometric function (of the first kind):


title: Beta distribution

text: is the rising factorial, also called the "Pochhammer symbol". The value of the characteristic function for "t" = 0, is one:


title: Beta distribution

text: Also, the real and imaginary parts of the characteristic function enjoy the following symmetries with respect to the origin of variable "t":


title: Beta distribution

text: The symmetric case α = β simplifies the characteristic function of the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometric function (of the first kind) reduces to a Bessel function (the modified Bessel function of the first kind formula_138 ) using Kummer's second transformation as follows:


title: Beta distribution

text: Another example of the symmetric case α = β = n/2 for beamforming applications can be found in Figure 11 of


title: Beta distribution

text: In the accompanying plots, the real part (Re) of the characteristic function of the beta distribution is displayed for symmetric (α = β) and skewed (α ≠ β) cases.


title: Beta distribution

text: multiplying the (exponential series) term formula_142 in the series of the moment generating function


title: Beta distribution

text: where ("x") is a Pochhammer symbol representing rising factorial. It can also be written in a recursive form as


title: Beta distribution

text: Since the moment generating function formula_145 has a positive radius of convergence, the beta distribution is determined by its moments.


title: Beta distribution

text: One can also show the following expectations for a transformed random variable, where the random variable "X" is Beta-distributed with parameters α and β: "X" ~ Beta(α, β). The expected value of the variable 1 − "X" is the mirror-symmetry of the expected value based on "X":


title: Beta distribution

text: Due to the mirror-symmetry of the probability density function of the beta distribution, the variances based on variables "X" and 1 − "X" are identical, and the covariance on "X"(1 − "X" is the negative of the variance:


title: Beta distribution

text: These are the expected values for inverted variables, (these are related to the harmonic means, see ):


title: Beta distribution

text: The following transformation by dividing the variable "X" by its mirror-image "X"/(1 − "X") results in the expected value of the "inverted beta distribution" or beta prime distribution (also known as beta distribution of the second kind or Pearson's Type VI):


title: Beta distribution

text: Variances of these transformed variables can be obtained by integration, as the expected values of the second moments centered on the corresponding variables:


title: Beta distribution

text: The following variance of the variable "X" divided by its mirror-image ("X"/(1−"X") results in the variance of the "inverted beta distribution" or beta prime distribution (also known as beta distribution of the second kind or Pearson's Type VI):


title: Beta distribution

text: Expected values for logarithmic transformations (useful for maximum likelihood estimates, see ) are discussed in this section. The following logarithmic linear transformations are related to the geometric means "G" and "G" (see ):


title: Beta distribution

text: Logit transformations are interesting, as they usually transform various shapes (including J-shapes) into (usually skewed) bell-shaped densities over the logit variable, and they may remove the end singularities over the original variable:


title: Beta distribution

text: Johnson considered the distribution of the logit - transformed variable ln("X"/1−"X"), including its moment generating function and approximations for large values of the shape parameters. This transformation extends the finite support [0, 1] based on the original variable "X" to infinite support in both directions of the real line (−∞, +∞).


title: Beta distribution

text: Higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two Gamma distributions and differentiating through the integral. They can be expressed in terms of higher order poly-gamma functions as follows:


title: Beta distribution

text: where the trigamma function, denoted ψ(α), is the second of the polygamma functions, and is defined as the derivative of the digamma function:


title: Beta distribution

text: The variances and covariance of the logarithmically transformed variables "X" and (1−"X") are different, in general, because the logarithmic transformation destroys the mirror-symmetry of the original variables "X" and (1−"X"), as the logarithm approaches negative infinity for the variable approaching zero.


title: Beta distribution

text: These logarithmic variances and covariance are the elements of the Fisher information matrix for the beta distribution. They are also a measure of the curvature of the log likelihood function (see section on Maximum likelihood estimation).


title: Beta distribution

text: Given a beta distributed random variable, "X" ~ Beta("α", "β"), the differential entropy of "X" is (measured in nats), the expected value of the negative of the logarithm of the probability density function:


title: Beta distribution

text: The digamma function "ψ" appears in the formula for the differential entropy as a consequence of Euler's integral formula for the harmonic numbers which follows from the integral:


title: Beta distribution

text: The differential entropy of the beta distribution is negative for all values of "α" and "β" greater than zero, except at "α" = "β" = 1 (for which values the beta distribution is the same as the uniform distribution), where the differential entropy reaches its maximum value of zero. It is to be expected that the maximum entropy should take place when the beta distribution becomes equal to the uniform distribution, since uncertainty is maximal when all possible events are equiprobable.


title: Beta distribution

text: For "α" or "β" approaching zero, the differential entropy approaches its minimum value of negative infinity. For (either or both) "α" or "β" approaching zero, there is a maximum amount of order: all the probability density is concentrated at the ends, and there is zero probability density at points located between the ends. Similarly for (either or both) "α" or "β" approaching infinity, the differential entropy approaches its minimum value of negative infinity, and a maximum amount of order. If either "α" or "β" approaches infinity (and the other is finite) all the probability density is concentrated at an end, and the probability density is zero everywhere else. If both shape parameters are equal (the symmetric case), "α" = "β", and they approach infinity simultaneously, the probability density becomes a spike (Dirac delta function) concentrated at the middle "x" = 1/2, and hence there is 100% probability at the middle "x" = 1/2 and zero probability everywhere else.


title: Beta distribution

text: The (continuous case) differential entropy was introduced by Shannon in his original paper (where he named it the "entropy of a continuous distribution"), as the concluding part of the same paper where he defined the discrete entropy. It is known since then that the differential entropy may differ from the infinitesimal limit of the discrete entropy by an infinite offset, therefore the differential entropy can be negative (as it is for the beta distribution). What really matters is the relative value of entropy.


title: Beta distribution

text: Given two beta distributed random variables, "X" ~ Beta("α", "β") and "X" ~ Beta("α"′, "β"′), the cross entropy is (measured in nats)


title: Beta distribution

text: The cross entropy has been used as an error metric to measure the distance between two hypotheses. Its absolute value is minimum when the two distributions are identical. It is the information measure most closely related to the log maximum likelihood (see section on "Parameter estimation. Maximum likelihood estimation")).


title: Beta distribution

text: The relative entropy, or Kullback–Leibler divergence "D"("X" || "X"), is a measure of the inefficiency of assuming that the distribution is "X" ~ Beta("α"′, "β"′) when the distribution is really "X" ~ Beta("α", "β"). It is defined as follows (measured in nats).


title: Beta distribution

text: The relative entropy, or Kullback–Leibler divergence, is always non-negative. A few numerical examples follow:


title: Beta distribution

text: The Kullback–Leibler divergence is not symmetric "D"("X" || "X") ≠ "D"("X" || "X") for the case in which the individual beta distributions Beta(1, 1) and Beta(3, 3) are symmetric, but have different entropies "h"("X") ≠ "h"("X"). The value of the Kullback divergence depends on the direction traveled: whether going from a higher (differential) entropy to a lower (differential) entropy or the other way around. In the numerical example above, the Kullback divergence measures the inefficiency of assuming that the distribution is (bell-shaped) Beta(3, 3), rather than (uniform) Beta(1, 1). The "h" entropy of Beta(1, 1) is higher than the "h" entropy of Beta(3, 3) because the uniform distribution Beta(1, 1) has a maximum amount of disorder. The Kullback divergence is more than two times higher (0.598803 instead of 0.267864) when measured in the direction of decreasing entropy: the direction that assumes that the (uniform) Beta(1, 1) distribution is (bell-shaped) Beta(3, 3) rather than the other way around. In this restricted sense, the Kullback divergence is consistent with the second law of thermodynamics.


title: Beta distribution

text: The Kullback–Leibler divergence is symmetric "D"("X" || "X") = "D"("X" || "X") for the skewed cases Beta(3, 0.5) and Beta(0.5, 3) that have equal differential entropy "h"("X") = "h"("X").


title: Beta distribution

text: follows from the above definitions and the mirror-symmetry "f"("x"; "α", "β") = "f"(1−"x"; "α", "β") enjoyed by the beta distribution.


title: Beta distribution

text: If 1 < α < β then mode ≤ median ≤ mean. Expressing the mode (only for α, β > 1), and the mean in terms of α and β:


title: Beta distribution

text: If 1 < β < α then the order of the inequalities are reversed. For α, β > 1 the absolute distance between the mean and the median is less than 5% of the distance between the maximum and minimum values of "x". On the other hand, the absolute distance between the mean and the mode can reach 50% of the distance between the maximum and minimum values of "x", for the (pathological) case of α = 1 and β = 1, for which values the beta distribution approaches the uniform distribution and the differential entropy approaches its maximum value, and hence maximum "disorder".


title: Beta distribution

text: It is known from the inequality of arithmetic and geometric means that the geometric mean is lower than the mean. Similarly, the harmonic mean is lower than the geometric mean. The accompanying plot shows that for α = β, both the mean and the median are exactly equal to 1/2, regardless of the value of α = β, and the mode is also equal to 1/2 for α = β > 1, however the geometric and harmonic means are lower than 1/2 and they only approach this value asymptotically as α = β → ∞.


title: Beta distribution

text: As remarked by Feller, in the Pearson system the beta probability density appears as type I (any difference between the beta distribution and Pearson's type I distribution is only superficial and it makes no difference for the following discussion regarding the relationship between kurtosis and skewness). Karl Pearson showed, in Plate 1 of his paper published in 1916, a graph with the kurtosis as the vertical axis (ordinate) and the square of the skewness as the horizontal axis (abscissa), in which a number of distributions were displayed. The region occupied by the beta distribution is bounded by the following two lines in the (skewness,kurtosis) plane, or the (skewness,excess kurtosis) plane:


title: Beta distribution

text: At a time when there were no powerful digital computers, Karl Pearson accurately computed further boundaries, for example, separating the "U-shaped" from the "J-shaped" distributions. The lower boundary line (excess kurtosis + 2 − skewness = 0) is produced by skewed "U-shaped" beta distributions with both values of shape parameters α and β close to zero. The upper boundary line (excess kurtosis − (3/2) skewness = 0) is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter. Karl Pearson showed that this upper boundary line (excess kurtosis − (3/2) skewness = 0) is also the intersection with Pearson's distribution III, which has unlimited support in one direction (towards positive infinity), and can be bell-shaped or J-shaped. His son, Egon Pearson, showed that the region (in the kurtosis/squared-skewness plane) occupied by the beta distribution (equivalently, Pearson's distribution I) as it approaches this boundary (excess kurtosis − (3/2) skewness = 0) is shared with the noncentral chi-squared distribution. Karl Pearson (Pearson 1895, pp. 357, 360, 373–376) also showed that the gamma distribution is a Pearson type III distribution. Hence this boundary line for Pearson's type III distribution is known as the gamma line. (This can be shown from the fact that the excess kurtosis of the gamma distribution is 6/"k" and the square of the skewness is 4/"k", hence (excess kurtosis − (3/2) skewness = 0) is identically satisfied by the gamma distribution regardless of the value of the parameter "k"). Pearson later noted that the chi-squared distribution is a special case of Pearson's type III and also shares this boundary line (as it is apparent from the fact that for the chi-squared distribution the excess kurtosis is 12/"k" and the square of the skewness is 8/"k", hence (excess kurtosis − (3/2) skewness = 0) is identically satisfied regardless of the value of the parameter "k"). This is to be expected, since the chi-squared distribution "X" ~ χ("k") is a special case of the gamma distribution, with parametrization X ~ Γ(k/2, 1/2) where k is a positive integer that specifies the "number of degrees of freedom" of the chi-squared distribution.


title: Beta distribution

text: An example of a beta distribution near the upper boundary (excess kurtosis − (3/2) skewness = 0) is given by α = 0.1, β = 1000, for which the ratio (excess kurtosis)/(skewness) = 1.49835 approaches the upper limit of 1.5 from below. An example of a beta distribution near the lower boundary (excess kurtosis + 2 − skewness = 0) is given by α= 0.0001, β = 0.1, for which values the expression (excess kurtosis + 2)/(skewness) = 1.01621 approaches the lower limit of 1 from above. In the infinitesimal limit for both α and β approaching zero symmetrically, the excess kurtosis reaches its minimum value at −2. This minimum value occurs at the point at which the lower boundary line intersects the vertical axis (ordinate). (However, in Pearson's original chart, the ordinate is kurtosis, instead of excess kurtosis, and it increases downwards rather than upwards).


title: Beta distribution

text: Values for the skewness and excess kurtosis below the lower boundary (excess kurtosis + 2 − skewness = 0) cannot occur for any distribution, and hence Karl Pearson appropriately called the region below this boundary the "impossible region". The boundary for this "impossible region" is determined by (symmetric or skewed) bimodal "U"-shaped distributions for which the parameters α and β approach zero and hence all the probability density is concentrated at the ends: "x" = 0, 1 with practically nothing in between them. Since for α ≈ β ≈ 0 the probability density is concentrated at the two ends "x" = 0 and "x" = 1, this "impossible boundary" is determined by a Bernoulli distribution, where the two only possible outcomes occur with respective probabilities "p" and "q" = 1−"p". For cases approaching this limit boundary with symmetry α = β, skewness ≈ 0, excess kurtosis ≈ −2 (this is the lowest excess kurtosis possible for any distribution), and the probabilities are "p" ≈ "q" ≈ 1/2. For cases approaching this limit boundary with skewness, excess kurtosis ≈ −2 + skewness, and the probability density is concentrated more at one end than the other end (with practically nothing in between), with probabilities formula_172 at the left end "x" = 0 and formula_173 at the right end "x" = 1.


title: Beta distribution

text: For certain values of the shape parameters α and β, the probability density function has inflection points, at which the curvature changes sign. The position of these inflection points can be useful as a measure of the dispersion or spread of the distribution.


title: Beta distribution

text: There are no inflection points in the remaining (symmetric and skewed) regions: U-shaped: (α, β < 1) upside-down-U-shaped: (1 < α < 2, 1 < β < 2), reverse-J-shaped (α < 1, β > 2) or J-shaped: (α > 2, β < 1)


title: Beta distribution

text: The accompanying plots show the inflection point locations (shown vertically, ranging from 0 to 1) versus α and β (the horizontal axes ranging from 0 to 5). There are large cuts at surfaces intersecting the lines α = 1, β = 1, α = 2, and β = 2 because at these values the beta distribution change from 2 modes, to 1 mode to no mode.


title: Beta distribution

text: The beta density function can take a wide variety of different shapes depending on the values of the two parameters "α" and "β". The ability of the beta distribution to take this great diversity of shapes (using only two parameters) is partly responsible for finding wide application for modeling actual measurements:


title: Beta distribution

text: The density function is skewed. An interchange of parameter values yields the mirror image (the reverse) of the initial curve, some more specific cases:


title: Beta distribution

text: Two unknown parameters (formula_240 of a beta distribution supported in the [0,1] interval) can be estimated, using the method of moments, with the first two moments (sample mean and sample variance) as follows. Let:


title: Beta distribution

text: When the distribution is required over a known interval other than [0, 1] with random variable "X", say ["a", "c"] with random variable "Y", then replace formula_247 with formula_248 and formula_249 with formula_250 in the above couple of equations for the shape parameters (see the "Alternative parametrizations, four parameters" section below)., where:


title: Beta distribution

text: All four parameters (formula_253 of a beta distribution supported in the ["a", "c"] interval -see section "Alternative parametrizations, Four parameters"-) can be estimated, using the method of moments developed by Karl Pearson, by equating sample and population values of the first four central moments (mean, variance, skewness and excess kurtosis). The excess kurtosis was expressed in terms of the square of the skewness, and the sample size ν = α + β, (see previous section "Kurtosis") as follows:


title: Beta distribution

text: One can use this equation to solve for the sample size ν= α + β in terms of the square of the skewness and the excess kurtosis as follows:


title: Beta distribution

text: This is the ratio (multiplied by a factor of 3) between the previously derived limit boundaries for the beta distribution in a space (as originally done by Karl Pearson) defined with coordinates of the square of the skewness in one axis and the excess kurtosis in the other axis (see ):


title: Beta distribution

text: The case of zero skewness, can be immediately solved because for zero skewness, α = β and hence ν = 2α = 2β, therefore α = β = ν/2


title: Beta distribution

text: (Excess kurtosis is negative for the beta distribution with zero skewness, ranging from -2 to 0, so that formula_259 -and therefore the sample shape parameters- is positive, ranging from zero when the shape parameters approach zero and the excess kurtosis approaches -2, to infinity when the shape parameters approach infinity and the excess kurtosis approaches zero).


title: Beta distribution

text: For non-zero sample skewness one needs to solve a system of two coupled equations. Since the skewness and the excess kurtosis are independent of the parameters formula_260, the parameters formula_261 can be uniquely determined from the sample skewness and the sample excess kurtosis, by solving the coupled equations with two known variables (sample skewness and sample excess kurtosis) and two unknowns (the shape parameters):


title: Beta distribution

text: Where one should take the solutions as follows: formula_267 for (negative) sample skewness < 0, and formula_268 for (positive) sample skewness > 0.


title: Beta distribution

text: The accompanying plot shows these two solutions as surfaces in a space with horizontal axes of (sample excess kurtosis) and (sample squared skewness) and the shape parameters as the vertical axis. The surfaces are constrained by the condition that the sample excess kurtosis must be bounded by the sample squared skewness as stipulated in the above equation. The two surfaces meet at the right edge defined by zero skewness. Along this right edge, both parameters are equal and the distribution is symmetric U-shaped for α = β < 1, uniform for α = β = 1, upside-down-U-shaped for 1 < α = β < 2 and bell-shaped for α = β > 2. The surfaces also meet at the front (lower) edge defined by "the impossible boundary" line (excess kurtosis + 2 - skewness = 0). Along this front (lower) boundary both shape parameters approach zero, and the probability density is concentrated more at one end than the other end (with practically nothing in between), with probabilities formula_269 at the left end "x" = 0 and formula_270 at the right end "x" = 1. The two surfaces become further apart towards the rear edge. At this rear edge the surface parameters are quite different from each other. As remarked, for example, by Bowman and Shenton, sampling in the neighborhood of the line (sample excess kurtosis - (3/2)(sample skewness) = 0) (the just-J-shaped portion of the rear edge where blue meets beige), "is dangerously near to chaos", because at that line the denominator of the expression above for the estimate ν = α + β becomes zero and hence ν approaches infinity as that line is approached. Bowman and Shenton write that "the higher moment parameters (kurtosis and skewness) are extremely fragile (near that line). However, the mean and standard deviation are fairly reliable." Therefore, the problem is for the case of four parameter estimation for very skewed distributions such that the excess kurtosis approaches (3/2) times the square of the skewness. This boundary line is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter. See for a numerical example and further comments about this rear edge boundary line (sample excess kurtosis - (3/2)(sample skewness) = 0). As remarked by Karl Pearson himself this issue may not be of much practical importance as this trouble arises only for very skewed J-shaped (or mirror-image J-shaped) distributions with very different values of shape parameters that are unlikely to occur much in practice). The usual skewed-bell-shape distributions that occur in practice do not have this parameter estimation problem.


title: Beta distribution

text: The remaining two parameters formula_260 can be determined using the sample mean and the sample variance using a variety of equations. One alternative is to calculate the support interval range formula_272 based on the sample variance and the sample kurtosis. For this purpose one can solve, in terms of the range formula_273, the equation expressing the excess kurtosis in terms of the sample variance, and the sample size ν (see and ):


title: Beta distribution

text: Another alternative is to calculate the support interval range formula_272 based on the sample variance and the sample skewness. For this purpose one can solve, in terms of the range formula_272, the equation expressing the squared skewness in terms of the sample variance, and the sample size ν (see section titled "Skewness" and "Alternative parametrizations, four parameters"):


title: Beta distribution

text: The remaining parameter can be determined from the sample mean and the previously obtained parameters: formula_280:


title: Beta distribution

text: The estimators "G" for sample skewness and "G" for sample kurtosis are used by DAP/SAS, PSPP/SPSS, and Excel. However, they are not used by BMDP and (according to ) they were not used by MINITAB in 1998. Actually, Joanes and Gill in their 1998 study concluded that the skewness and kurtosis estimators used in BMDP and in MINITAB (at that time) had smaller variance and mean-squared error in normal samples, but the skewness and kurtosis estimators used in DAP/SAS, PSPP/SPSS, namely "G" and "G", had smaller mean-squared error in samples from a very skewed distribution. It is for this reason that we have spelled out "sample skewness", etc., in the above formulas, to make it explicit that the user should choose the best estimator according to the problem at hand, as the best estimator for skewness and kurtosis depends on the amount of skewness (as shown by Joanes and Gill).


title: Beta distribution

text: As is also the case for maximum likelihood estimates for the gamma distribution, the maximum likelihood estimates for the beta distribution do not have a general closed form solution for arbitrary values of the shape parameters. If "X", ..., "X" are independent random variables each having a beta distribution, the joint log likelihood function for "N" iid observations is:


title: Beta distribution

text: Finding the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal to zero yielding the maximum likelihood estimator of the shape parameters:


title: Beta distribution

text: since the digamma function denoted ψ(α) is defined as the logarithmic derivative of the gamma function:


title: Beta distribution

text: To ensure that the values with zero tangent slope are indeed a maximum (instead of a saddle-point or a minimum) one has to also satisfy the condition that the curvature is negative. This amounts to satisfying that the second partial derivative with respect to the shape parameters is negative


title: Beta distribution

text: where the trigamma function, denoted "ψ"("α"), is the second of the polygamma functions, and is defined as the derivative of the digamma function:


title: Beta distribution

text: These conditions are equivalent to stating that the variances of the logarithmically transformed variables are positive, since:


title: Beta distribution

text: Alternatively, the condition of negative curvature at a maximum is also equivalent to stating that the following logarithmic derivatives of the geometric means "G" and "G" are positive, since:


title: Beta distribution

text: From the condition that at a maximum, the partial derivative with respect to the shape parameter equals zero, we obtain the following system of coupled maximum likelihood estimate equations (for the average log-likelihoods) that needs to be inverted to obtain the (unknown) shape parameter estimates formula_302 in terms of the (known) average of logarithms of the samples "X", ..., "X":


title: Beta distribution

text: where we recognize formula_304 as the logarithm of the sample geometric mean and formula_305 as the logarithm of the sample geometric mean based on (1 − "X"), the mirror-image of "X". For formula_306, it follows that formula_307.


title: Beta distribution

text: These coupled equations containing digamma functions of the shape parameter estimates formula_302 must be solved by numerical methods as done, for example, by Beckman et al. Gnanadesikan et al. give numerical solutions for a few cases. N.L.Johnson and S.Kotz suggest that for "not too small" shape parameter estimates formula_302, the logarithmic approximation to the digamma function formula_311 may be used to obtain initial values for an iterative solution, since the equations resulting from this approximation can be solved exactly:


title: Beta distribution

text: which leads to the following solution for the initial values (of the estimate shape parameters in terms of the sample geometric means) for an iterative solution:


title: Beta distribution

text: Alternatively, the estimates provided by the method of moments can instead be used as initial values for an iterative solution of the maximum likelihood coupled equations in terms of the digamma functions.


title: Beta distribution

text: When the distribution is required over a known interval other than [0, 1] with random variable "X", say ["a", "c"] with random variable "Y", then replace ln("X") in the first equation with


title: Beta distribution

text: If one of the shape parameters is known, the problem is considerably simplified. The following logit transformation can be used to solve for the unknown shape parameter (for skewed cases such that formula_318, otherwise, if symmetric, both -equal- parameters are known when one is known):


title: Beta distribution

text: This logit transformation is the logarithm of the transformation that divides the variable "X" by its mirror-image ("X"/(1 - "X") resulting in the "inverted beta distribution" or beta prime distribution (also known as beta distribution of the second kind or Pearson's Type VI) with support [0, +∞). As previously discussed in the section "Moments of logarithmically transformed random variables," the logit transformation formula_320, studied by Johnson, extends the finite support [0, 1] based on the original variable "X" to infinite support in both directions of the real line (−∞, +∞).


title: Beta distribution

text: If, for example, formula_321 is known, the unknown parameter formula_322 can be obtained in terms of the inverse digamma function of the right hand side of this equation:


title: Beta distribution

text: In particular, if one of the shape parameters has a value of unity, for example for formula_325 (the power function distribution with bounded support [0,1]), using the identity ψ("x" + 1) = ψ("x") + 1/"x" in the equation formula_326, the maximum likelihood estimator for the unknown parameter formula_322 is, exactly: