|Gaussian integer factorization|
|Roman numeral (unicode)||Ⅱ, ⅱ|
|Arabic & Kurdish||٢|
An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal, hexadecimal, or in any other base that is even, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.
Two is the smallest prime number, and the only even prime number (for this reason it is sometimes called "the oddest prime"). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Ramanujan prime.
Two is the third (or fourth) Fibonacci number.
Two is the base of the binary system, the numeral system with the least number of tokens allowing to denote a natural number n substantially more concise (log2 n tokens), compared to a direct representation by the corresponding count of a single token (n tokens). This binary number system is used extensively in computing.
For any number x:
Extending this sequence of operations by introducing the notion of hyperoperations, here denoted by "hyper(a,b,c)" with a and c being the first and second operand, and b being the level in the above sketched sequence of operations, the following holds in general:
Two has therefore the unique property that 2 + 2 = 2 · 2 = 22 = 2↑↑2 = 2↑↑↑2 = ..., disregarding the level of the hyperoperation, here denoted by Knuth's up-arrow notation. The number of up-arrows refers to the level of the hyperoperation.
Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols
This comes from the fact that:
Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic and other roots, may simply be left blank for square roots, as it is tacitly understood.
The smallest field has two elements.
Two also has the unique property such that
for a not equal to zero
|2 × x||2||4||6||8||10||12||14||16||18||20||22||24||26||28||30||32||34||36||38||40||42||44||46||48||50||100||200||2000|
|2 ÷ x||2||1||0.6||0.5||0.4||0.3||0.285714||0.25||0.2||0.2||0.18||0.16||0.153846||0.142857||0.13|
|x ÷ 2||0.5||1||1.5||2||2.5||3||3.5||4||4.5||5||5.5||6||6.5||7||7.5|
The glyph used in the modern Western world to represent the number 2 traces its roots back to the Indic Brahmic script, where "2" was written as two horizontal lines. The modern Chinese and Japanese languages still use this method. The Gupta script rotated the two lines 45 degrees, making them diagonal. The top line was sometimes also shortened and had its bottom end curve towards the center of the bottom line. In the Nagari script, the top line was written more like a curve connecting to the bottom line. In the Arabic Ghubar writing, the bottom line was completely vertical, and the glyph looked like a dotless closing question mark. Restoring the bottom line to its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our modern glyph.
The number 2 is important in Judaism, with one of the earliest references being that God ordered Noah to put two of every unclean animal (Gen. 7:2) in his ark (see Noah's Ark). Later on, the Ten Commandments were given in the form of two tablets. The number also has ceremonial importance, such as the two candles that are traditionally kindled to usher in the Shabbat, recalling the two different ways Shabbat is referred to in the two times the Ten Commandments are recorded in the Torah. These two expressions are known in Hebrew as שמור וזכור ("guard" and "remember"), as in "Guard the Shabbat day to sanctify it" (Deut. 5:12) and "Remember the Shabbat day to sanctify it" (Ex. 20:8). Two challot (lechem mishneh) are placed on the table for each Shabbat meal and a blessing made over them, to commemorate the double portion of manna which fell in the desert every Friday to cover that day's meals and the Shabbat meals.
"Second-Day Yom Tov" (Yom Tov Sheini Shebegaliyot) is a rabbinical enactment that mandates a two-day celebration for each of the one-day Jewish festivals (i.e., the first and seventh day of Passover, the day of Shavuot, the first day of Sukkot, and the day of Shemini Atzeret) outside the Land of Israel.
The most common philosophical dichotomy is perhaps the one of good and evil, but there are many others. See dualism for an overview. In Hegelian dialectic, the process of synthesis reconciles two different perspectives into one.
The ancient Sanskrit language of India, does not only have a singular and plural form for nouns, as do many other languages, but instead has, a singular (1) form, a dual (2) form, and a plural (everything above 2) form, for all nouns, due to the significance of 2. It is viewed as important because of the anatomical significance of 2 (2 hands, 2 nostrils, 2 eyes, 2 legs, etc.)
Two (二, èr) is a good number in Chinese culture. There is a Chinese saying, "good things come in pairs". It is common to use double symbols in product brand names, e.g. double happiness, double coin, double elephants etc. Cantonese people like the number two because it sounds the same as the word "easy" (易) in Cantonese.
The 2008–09 UEFA Champions League was the 54th edition of Europe's premier club football tournament and the 17th edition under the current UEFA Champions League format. The final was played at the Stadio Olimpico in Rome on 27 May 2009. It was the eighth time the European Cup final has been held in Italy and the fourth time it has been held at the Stadio Olimpico. The final was contested by the defending champions, Manchester United, and Barcelona, who had last won the tournament in 2006. Barcelona won the match 2–0, with goals from Samuel Eto'o and Lionel Messi, securing The Treble in the process. In addition, both UEFA Cup finalists, Werder Bremen and Shakhtar Donetsk featured in the Champions League group stage.
Anorthosis of Cyprus and BATE Borisov of Belarus were the first teams from their respective countries to qualify for the group stage. Romanian side CFR Cluj and Russian champions Zenit Saint Petersburg also made their Champions League debuts.2013–14 UEFA Champions League
The 2013–14 UEFA Champions League was the 59th season of Europe's premier club football tournament organised by UEFA, and the 22nd season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.
The 2014 UEFA Champions League Final was played between Real Madrid and Atlético Madrid at the Estádio da Luz in Lisbon, Portugal, marking the fifth final to feature two teams from the same association, and the first time in tournament history that both finalists are from the same city. Real Madrid, who eliminated the title holders, Bayern Munich, in the semi-finals, won in extra time, giving them a record-extending tenth European Cup/Champions League title. Real equalized late in the second half through Sergio Ramos and then pulled away during extra time to win 4–1.For the first time, the clubs who qualified for the group stage also qualified for the newly formed 2013–14 UEFA Youth League, a competition available to players aged 19 or under.2014–15 UEFA Champions League
The 2014–15 UEFA Champions League was the 60th season of Europe's premier club football tournament organised by UEFA, and the 23rd season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.
The 2015 UEFA Champions League Final was played at the Olympiastadion in Berlin, Germany, with Spanish side Barcelona defeating Italian side Juventus by 3–1 to win their fifth title and complete their treble. Real Madrid were the title holders, but they were eliminated by Juventus in the semi-finals.
This season was the first where clubs must comply with UEFA Financial Fair Play Regulations in order to participate. Moreover, this season was the first where a club from Gibraltar competed in the tournament, after the Gibraltar Football Association was accepted as the 54th UEFA member at the UEFA Congress in May 2013. They were granted one spot in the Champions League, which was taken by Lincoln Red Imps, the champions of the 2013–14 Gibraltar Premier Division.On 17 July 2014, the UEFA emergency panel ruled that Ukrainian and Russian clubs would not be drawn against each other "until further notice" due to the political unrest between the countries. Another ruling centred in regional instability was also made where Israeli teams were prohibited from hosting any UEFA competitions due to the 2014 Israel–Gaza conflict. The rules regarding suspension due to yellow card accumulation were also changed such that all bookings expired on completion of the quarter-finals and were not carried forward to the semi-finals. Moreover, this was the first season in which vanishing spray was used.2015–16 UEFA Champions League
The 2015–16 UEFA Champions League was the 61st season of Europe's premier club football tournament organised by UEFA, and the 24th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League. Barcelona were the title holders, but were eliminated by Atlético Madrid in the quarter-finals.
The 2016 UEFA Champions League Final was played between Real Madrid and Atlético Madrid at the San Siro in Milan, Italy. It was the second time in the tournament's history that both finalists were from the same city, after the same clubs faced each other in the 2014 final. Real Madrid defeated Atlético Madrid 5–3 on penalties (1–1 after extra time) in the final to win a record-extending eleventh European Cup/Champions League title.
As the winners of the 2015–16 UEFA Champions League, Real Madrid qualified as the UEFA representative at the 2016 FIFA Club World Cup in Japan (their third Club World Cup appearance), and also earned the right to play against the winners of the 2015–16 UEFA Europa League, Sevilla, in the 2016 UEFA Super Cup.2016–17 UEFA Champions League
The 2016–17 UEFA Champions League was the 62nd season of Europe's premier club football tournament organised by UEFA, and the 25th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.
The final was played between Juventus and Real Madrid at the Millennium Stadium in Cardiff, Wales. It was the second time that the two teams faced each other in the competition's decisive match, having previously met in the 1998 final. Real Madrid, the defending champions, beat Juventus 4–1 to win a record-extending 12th title. With this victory, Real Madrid became the first team to successfully defend their title in the Champions League era, and the first to do so since Milan in 1990.
As winners, Real Madrid qualified as the UEFA representative for the 2017 FIFA Club World Cup in the United Arab Emirates, and also earned the right to play against the winners of the 2016–17 UEFA Europa League, Manchester United, in the 2017 UEFA Super Cup.2017–18 UEFA Champions League
The 2017–18 UEFA Champions League was the 63rd season of Europe's premier club football tournament organised by UEFA, the 26th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.
The final was played between Real Madrid and Liverpool at the NSC Olimpiyskiy Stadium in Kiev, Ukraine. Real Madrid, the defending champions, beat Liverpool 3–1 to win a record-extending 13th title and their third title in a row.
As winners, Real Madrid qualified as the UEFA representative for the 2018 FIFA Club World Cup in the United Arab Emirates, and also earned the right to play against the winners of the 2017–18 UEFA Europa League, Atlético Madrid, in the 2018 UEFA Super Cup. Moreover, they would also have been automatically qualified for the 2018–19 UEFA Champions League group stage, but since they had already qualified through their league performance, the berth reserved was given to the champions of the 2017–18 Czech First League, the 11th-ranked association according to next season's access list.2018–19 Premier League
The 2018–19 Premier League is the 27th season of the Premier League, the top English professional league for association football clubs, since its establishment in 1992. The season started on 10 August 2018 and is scheduled to finish on 12 May 2019. Fixtures for the 2018–19 season were announced on 14 June 2018.Manchester City are the defending champions. Wolverhampton Wanderers, Cardiff City and Fulham joined as the promoted clubs from the 2017–18 EFL Championship. They replaced West Bromwich Albion, Swansea City and Stoke City who were relegated to the 2018–19 EFL Championship.
The season saw the occurrence of two aviation incidents involving Premier League personnel. On 27 October 2018, Leicester City owner Vichai Srivaddhanaprabha was killed in a helicopter crash outside the King Power Stadium, shortly after a 1–1 home draw against West Ham United. Almost three months later, on 21 January 2019, Cardiff City player Emiliano Sala, en route to join the club following his record signing from Nantes, died on board a Piper PA-46 Malibu aircraft that crashed off Alderney.2018–19 UEFA Champions League
The 2018–19 UEFA Champions League is the 64th season of Europe's premier club football tournament organised by UEFA, and the 27th season since it was renamed from the European Champion Clubs' Cup to the UEFA Champions League.
The final will be played at the Wanda Metropolitano in Madrid, Spain. The winners of the 2018–19 UEFA Champions League will earn the right to play against the winners of the 2018–19 UEFA Europa League in the 2019 UEFA Super Cup. They will also automatically qualify for the 2019–20 UEFA Champions League group stage, and if they have already qualified through their league performance, the berth reserved will be given to the champions of the 2018–19 Austrian Bundesliga, the 11th-ranked association according to next season's access list.For the first time, the video assistant referee (VAR) system was used in the competition from the round of 16 onward.Real Madrid were the defending champions, having won the title for three successive seasons in 2015–16, 2016–17 and 2017–18. However, they were eliminated by Ajax in the round of 16.2018–19 UEFA Europa League
The 2018–19 UEFA Europa League is the 48th season of Europe's secondary club football tournament organised by UEFA, and the 10th season since it was renamed from the UEFA Cup to the UEFA Europa League.
The final will be played at the Olympic Stadium in Baku, Azerbaijan. The winners of the 2018–19 UEFA Europa League will earn the right to play against the winners of the 2018–19 UEFA Champions League in the 2019 UEFA Super Cup. They will also automatically qualify for the 2019–20 UEFA Champions League group stage, and if they have already qualified through their league performance, the berth reserved will be given to the third-placed team of the 2018–19 Ligue 1, the 5th-ranked association according to next season's access list.For the first time, the video assistant referee (VAR) system will be used in the competition, where it will be implemented in the final.As the title holders of Europa League, Atlético Madrid qualified for the 2018–19 UEFA Champions League, although they had already qualified before the final through their league performance. They are unable to defend their title as they advanced to the Champions League knockout stage.2 Chainz
Tauheed Epps (born September 12, 1977), known professionally as 2 Chainz (formerly Tity Boi), is an American rapper, media personality and basketball player. Born and raised in College Park, Georgia, he initially gained recognition for being one-half of the Southern hip hop duo Playaz Circle, alongside his longtime friend and fellow rapper Earl "Dolla Boy" Conyers. The duo was signed to fellow Georgia-based rapper Ludacris' Disturbing tha Peace label, and are best known for their debut single "Duffle Bag Boy".
In February 2012, Epps signed a solo record deal with Def Jam Recordings, an imprint of Universal Music Group. The following August, he released his debut studio album Based on a T.R.U. Story, to mixed reviews. The album spawned three successful singles: "No Lie," "Birthday Song," and "I'm Different", all of which charted in the top 50 of the Billboard Hot 100 and were certified Gold or higher by the RIAA, along with the album being certified Gold. His second studio album B.O.A.T.S. II: Me Time was released on September 11, 2013; supported by the lead singles "Feds Watching" and "Used 2". Epps began working with the TV Network Viceland on a show called Most Expensivest, which debuted its first episode on November 15, 2017, and is currently in its second season.Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
for n > 1.
One has F2 = 1. In some books, and particularly in old ones, F0, the "0" is omitted, and the Fibonacci sequence starts with F1 = F2 = 1. The beginning of the sequence is thus:
Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.
Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. They appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.
Fibonacci numbers are also closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . Lucas numbers are also intimately connected with the golden ratio.Golden Globe Award
The Golden Globe Awards are accolades bestowed by the 93 members of the Hollywood Foreign Press Association beginning in January 1944, recognizing excellence in film and television, both domestic and foreign.
The annual ceremony at which the awards are presented is a major part of the film industry's awards season, which culminates each year in the Academy Awards. The eligibility period for the Golden Globes corresponds to the calendar year (i.e. January 1 through December 31). The 76th Golden Globe Awards, honoring the best in film and television in 2018, were held on January 6, 2019. The 77th Golden Globe Awards will take place on January 5, 2020.Guardians of the Galaxy Vol. 2
Guardians of the Galaxy Vol. 2 is a 2017 American superhero film based on the Marvel Comics superhero team Guardians of the Galaxy, produced by Marvel Studios and distributed by Walt Disney Studios Motion Pictures. It is the sequel to 2014's Guardians of the Galaxy and the fifteenth film in the Marvel Cinematic Universe (MCU). Written and directed by James Gunn, the film stars an ensemble cast featuring Chris Pratt, Zoe Saldana, Dave Bautista, Vin Diesel, Bradley Cooper, Michael Rooker, Karen Gillan, Pom Klementieff, Elizabeth Debicki, Chris Sullivan, Sean Gunn, Sylvester Stallone, and Kurt Russell. In Guardians of the Galaxy Vol. 2, the Guardians travel throughout the cosmos as they help Peter Quill learn more about his mysterious parentage.
The film was officially announced at the 2014 San Diego Comic-Con International before the theatrical release of the first film, along with James Gunn's return from the first film, with the title of the sequel revealed a year later in June 2015. Principal photography began in February 2016 at Pinewood Atlanta Studios in Fayette County, Georgia, with many crew changes from the first film due to other commitments. Filming concluded in June 2016. James Gunn chose to set the sequel shortly after the first film to explore the characters' new roles as the Guardians, and to follow the storyline of Quill's father established throughout that previous film. Russell was confirmed as Quill's father in July 2016, portraying Ego, a departure from Quill's comic father.
Guardians of the Galaxy Vol. 2 premiered in Tokyo on April 10, 2017 and was released in the United States on May 5, in 3D and IMAX 3D. It grossed more than $863 million worldwide, making it the eighth-highest-grossing film of 2017, while also outgrossing its predecessor. The film received praise for its visuals, soundtrack, humor, and cast, though some critics deemed it inferior to the original. It received a nomination for Best Visual Effects at the 90th Academy Awards. A sequel, Guardians of the Galaxy Vol. 3 is in development, with Gunn returning as writer and director.List of NBA champions
The National Basketball Association (NBA) (formerly Basketball Association of America (BAA) from 1946 to 1949) Finals is the championship series for the NBA and the conclusion of the NBA's postseason. All Finals have been played in a best-of-seven format, and contested between the winners of the Eastern Conference and the Western Conference (formerly Divisions before 1970), except in 1950 in which the Eastern Division champion faced the winner between the Western and Central Division champions. Prior to 1949, the playoffs were instituted a three-stage tournament where the two semifinal winners played each other in the finals. The winning team of the series receives the Larry O'Brien Championship Trophy.
The home-and-away format in the NBA Finals is in a 2–2–1–1–1 format (the team with the better regular season record plays on their home court in Games 1, 2, 5 and 7) during 1947–1948, 1950–1952, 1957–1970, 1972–1974, 1976–1977, 1979–1984, 2014–present. It was previously in a 2–3–2 format (the team with the better regular season record plays on their home court in Games 1, 2, 6 and 7) during 1949, 1953–1955, and 1985–2013, in a 1–1–1–1–1–1–1 format during 1956 and 1971 and in a 1–2–2–1–1 format during 1975 and 1978.The Eastern Conference/Division leads the Western Conference/Division in series won (38–32). The defunct Central Division won one championship. The Boston Celtics and the Minneapolis/Los Angeles Lakers alone own almost half of the titles, having won a combined 33 of 72 championships.Normal distribution
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).
The probability density of the normal distribution is
whereNorthrop Grumman B-2 Spirit
The Northrop (later Northrop Grumman) B-2 Spirit, also known as the Stealth Bomber, is an American heavy strategic bomber, featuring low observable stealth technology designed for penetrating dense anti-aircraft defenses; it is a flying wing design with a crew of two. The bomber can deploy both conventional and thermonuclear weapons, such as eighty 500-pound class (230 kg) Mk 82 JDAM Global Positioning System-guided bombs, or sixteen 2,400-pound (1,100 kg) B83 nuclear bombs. The B-2 is the only acknowledged aircraft that can carry large air-to-surface standoff weapons in a stealth configuration.
Development started under the "Advanced Technology Bomber" (ATB) project during the Carter administration; its expected performance was one of his reasons for the cancellation of the supersonic B-1A bomber. The ATB project continued during the Reagan administration, but worries about delays in its introduction led to the reinstatement of the B-1 program. Program costs rose throughout development. Designed and manufactured by Northrop, later Northrop Grumman, the cost of each aircraft averaged US$737 million (in 1997 dollars). Total procurement costs averaged $929 million per aircraft, which includes spare parts, equipment, retrofitting, and software support. The total program cost, which included development, engineering and testing, averaged $2.1 billion per aircraft in 1997.Because of its considerable capital and operating costs, the project was controversial in the U.S. Congress. The winding-down of the Cold War in the latter portion of the 1980s dramatically reduced the need for the aircraft, which was designed with the intention of penetrating Soviet airspace and attacking high-value targets. During the late 1980s and 1990s, Congress slashed plans to purchase 132 bombers to 21. In 2008, a B-2 was destroyed in a crash shortly after takeoff, though the crew ejected safely. Twenty B-2s are in service with the United States Air Force, which plans to operate them until 2032.The B-2 is capable of all-altitude attack missions up to 50,000 feet (15,000 m), with a range of more than 6,000 nautical miles (6,900 mi; 11,000 km) on internal fuel and over 10,000 nautical miles (12,000 mi; 19,000 km) with one midair refueling. It entered service in 1997 as the second aircraft designed to have advanced stealth technology after the Lockheed F-117 Nighthawk attack aircraft. Though designed originally as primarily a nuclear bomber, the B-2 was first used in combat dropping conventional, non-nuclear ordnance in the Kosovo War in 1999. It later served in Iraq, Afghanistan, and Libya.PlayStation 2
The PlayStation 2 (PS2) is a home video game console that was developed by Sony Computer Entertainment. It is the successor to the original PlayStation console and is the second iteration in the PlayStation lineup of consoles. It was released in 2000 and competed with Sega's Dreamcast, Nintendo's GameCube and Microsoft's Xbox in the sixth generation of video game consoles.
Announced in 1999, the PlayStation 2 offered backwards compatibility for its predecessor's DualShock controller, as well as for its games. The PlayStation 2 is the best-selling video game console of all time, selling over 155 million units, with 150 million confirmed by Sony in 2011. More than 3,874 game titles have been released for the PS2 since launch, and more than 1.5 billion copies have been sold. Sony later manufactured several smaller, lighter revisions of the console known as Slimline models in 2004. In 2006, Sony announced and launched its successor, the PlayStation 3.
Even with the release of its successor, the PlayStation 2 remained popular well into the seventh generation and continued to be produced until January 4, 2013, when Sony finally announced that the PlayStation 2 had been discontinued after 12 years of production – one of the longest runs for a video game console. Despite the announcement, new games for the console continued to be produced until the end of 2013, including Final Fantasy XI: Seekers of Adoulin for Japan, FIFA 13 for North America, and Pro Evolution Soccer 2014 for Europe. Repair services for the system in Japan ended on September 7, 2018.Two and a Half Men
Two and a Half Men is an American television sitcom that originally aired on CBS for twelve seasons from September 22, 2003, to February 19, 2015. Originally starring Charlie Sheen, Jon Cryer, and Angus T. Jones, the series was about a hedonistic jingle writer, Charlie Harper, his uptight brother, Alan, and Alan's troublesome son, Jake. After Alan divorces, he and Jake move into Charlie's beachfront Malibu house and complicate Charlie's freewheeling life.
In 2010, CBS and Warner Bros. Television reached a multiyear broadcasting agreement for the series, renewing it through at least the 2011–12 season. In February 2011, however, CBS and Warner Bros. decided to end production for the rest of the eighth season after Sheen entered drug rehabilitation and made "disparaging comments" about the series' creator and executive producer Chuck Lorre. Sheen's contract was terminated the following month and he was confirmed not to be returning to the series. Ashton Kutcher was hired to replace him the following season as Walden Schmidt, a billionaire who buys Charlie's house after his death.
In April 2013, CBS renewed the series for an eleventh season after closing one-year deals with Kutcher and Cryer. Jones, who was attending college, was relegated to recurring status for season 11 but did not make an appearance until the series finale. He was replaced by Jenny (Amber Tamblyn), Charlie's previously unknown daughter. In March 2014, CBS renewed the series for a twelfth season, which was later announced to be the series' last. The season began airing in October 2014 and concluded in February 2015 with the 40-minute series finale "Of Course He's Dead". The success of the series led to it being the fourth-highest revenue-generating program for 2012, earning $3.24 million an episode.Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , or .