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Don't Bring Me Down
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This article is about the Electric Light Orchestra song. For other uses, see Don't Bring Me Down (disambiguation).
"Don't Bring Me Down"
Artwork for UK, Australian, and some other European vinyl releases
Single by Electric Light Orchestra
from the album Discovery
B-side
"Dreaming of 4000"
Released
1979
Format
7" single
Genre
Pop rock[1]
Length
4:02
Label
Jet
Songwriter(s)
Jeff Lynne
Producer(s)
Jeff Lynne
Electric Light Orchestra singles chronology
"The Diary of Horace Wimp"
(1979) "Don't Bring Me Down"
(1979) "Confusion/Last Train to London"
(1979)
"The Diary of Horace Wimp"
(1979) "Don't Bring Me Down"
(1979) "Confusion/Last Train to London"
(1979)
Audio sample
file
help
Discovery track listing
9 tracks [show]
Side one
"Shine a Little Love"
"Confusion"
"Need Her Love"
"The Diary of Horace Wimp"
Side two
"Last Train to London"
"Midnight Blue"
"On the Run"
"Wishing"
"Don't Bring Me Down"
Music video
"Don't Bring Me Down" on YouTube
"Don't Bring Me Down" is the ninth and final track on the English rock band the Electric Light Orchestra's 1979 album Discovery. It is their highest charting hit in the United States to date.
Contents
[hide]
1 History
2 Misheard lyric
3 Music video
4 Jeff Lynne version
5 Personnel
6 Cover versions, samplings and remixes
7 Chart and sales
7.1 Weekly charts
7.2 Year-end charts
7.3 Sales and certifications
8 See also
9 References
10 External links
History[edit]
“ It's a great big galloping ball of distortion. I wrote it at the last minute, 'cos I felt there weren't enough loud ones on the album. This was just what I was after. ”
— Discovery remaster (2001), Jeff Lynne
"Don't Bring Me Down" is the band's second highest charting hit in the UK where it peaked at number 3[2] and their biggest hit in the United States, peaking at number 4 on the Billboard Hot 100.[3] It also charted well in Canada (number 1) and Australia (number 6). This was the first song by ELO not to include a string section.[4]
The drum track is in fact a tape loop, coming from "On the Run" looped and slowed down.[4]
The song ends with the sound of a door slamming. According to producer Jeff Lynne, this was a metal fire door at Musicland Studios where the song was recorded.[4]
The song was dedicated to the NASA Skylab space station, which re-entered the Earth's atmosphere and burned up over the Indian Ocean and Western Australia on 11 July 1979.[4]
On 4 November 2007, Lynne was awarded a BMI (Broadcast Music, Inc) Million-Air certificate for "Don't Bring Me Down" for the song having reached two million airplays. An instrumental version was also music for the interludes in the computer game Frantic Freddie.
Misheard lyric[edit]
A common mondegreen in the song is the perception that, following the title line, Lynne shouts "Bruce!" In the liner notes of the ELO compilation Flashback and elsewhere, Lynne has explained that he is singing a made-up word, "Grooss," which some have suggested sounds like the German expression "Gruß." After the song's release, so many people had misinterpreted the word as "Bruce" that Lynne actually began to sing the word as "Bruce" for fun at live shows.[5][6]
Music video[edit]
A music video for the song was produced, which showed video of the band performing the song interspersed with various animations relating to the song's subject matter, including big-bottomed majorettes and a pulsating neon frankfurter. The band's three resident string players are depicted playing keyboards in the music video.
Jeff Lynne version[edit]
"Don't Bring Me Down"
Song by Electric Light Orchestra
from the album Mr. Blue Sky: The Very Best of Electric Light Orchestra
Released
8 October 2012
Recorded
2001–2012 Bungalow Palace
Length
4:01
Label
Frontiers
Songwriter(s)
Jeff Lynne
Producer(s)
Jeff Lynne
Mr. Blue Sky: The Very Best of Electric Light Orchestra track listing
12 tracks [show]
"Mr. Blue Sky"
"Evil Woman"
"Strange Magic"
"Don't Bring Me Down"
"Turn to Stone"
"Showdown"
"Telephone Line"
"Livin' Thing"
"Do Ya"
"Can't Get It Out of My Head"
"10538 Overture"
"Point of No Return"
Jeff Lynne re-recorded the song in his own home studio. It was released in a compilation album with other re-recorded ELO songs and under the ELO name called Mr. Blue Sky: The Very Best of Electric Light Orchestra.[7]
Personnel[edit]
Jeff Lynne – vocals, guitar, piano, synthesizer[4]
Bev Bevan – drums, percussion[4]
Richard Tandy – grand piano, synthesizer, electric piano, clavinet[4]
Kelly Groucutt – bass, vocals[4]
Cover versions, samplings and remixes[edit]
In 2012, The Hives released a song called "Go Right Ahead". Though not a direct cover, the main riff in the song is nearly identical to the one in "Don't Bring Me Down",[8] and as a result Jeff Lynne was officially credited as a co-writer.
Chart and sales[edit]
Weekly charts[edit]
Chart (1979)
Peak
position
Australia (Kent Music Report)[9] 6
Austria (Ö3 Austria Top 40)[10] 2
Belgium (Ultratop 50 Flanders)[11] 5
Canada Top Singles (RPM)[12] 1
France (IFOP)[13] 44
Germany (Official German Charts)[14] 5
Ireland (IRMA)[15] 6
Netherlands (Dutch Top 40)[16] 5
Netherlands (Single Top 100)[17] 5
New Zealand (Recorded Music NZ)[18] 6
South Africa (Springbok Radio)[19] 9
Spain (AFYVE)[20] 10
Switzerland (Schweizer Hitparade)[21] 2
UK Singles (Official Charts Company)[2] 3
US Billboard Hot 100[3] 4
US Cash Box[22] 4
US Record World[23] 3
Year-end charts[edit]
Chart (1979)
Rank
Australia (Kent Music Report)[24] 34
Belgium (Ultratop 50 Flanders)[25] 57
Canada Top Singles (RPM)[26] 25
Netherlands (Dutch Top 40)[27] 39
Netherlands (Single Top 100)[28] 61
New Zealand (Recorded Music NZ)[29] 50
US Billboard Hot 100[30] 81
US Cash Box[31] 39
Sales and certifications[edit]
Region
Certification
Certified units/Sales
United Kingdom (BPI)[32]
Silver 250,000^
United States (RIAA)[33]
Gold 1,000,000^
*sales figures based on certification alone
^shipments figures based on certification alone
See also[edit]
List of number-one singles of 1979 (Canada)
References[edit]
Jump up ^ Guarisco, Donald A. "Don't Bring Me Down – Song Review". AllMusic. All Media Network. Retrieved 26 May 2013.
^ Jump up to: a b "Official Singles Chart Top 100". Official Charts Company. Retrieved 5 March 2013.
^ Jump up to: a b "Electric Light Orchestra – Awards". AllMusic. All Media Network. Retrieved 5 March 2013.
^ Jump up to: a b c d e f g h Porter, Robert. "Electric Light Orchestra and Jeff Lynne -- Don't Bring Me Down: An in-depth song analysis". Jefflynnesongs.com. Retrieved 26 May 2013.
Jump up ^ Wild, David. "The Story of a Rock and Roll Band and the Pop Genius Who Dared to Go Baroque". Flashback (Media notes).
Jump up ^ Sullivan, Caroline (15 October 2014). "ELO's Jeff Lynne: 'All those hipsters with beards are copying me!'". The Guardian. Retrieved 23 August 2016.
Jump up ^ "Mr. Blue Sky – The Very Best of Electric Light Orchestra". Archived from the original on 27 October 2015. . Jefflynneselo.com.
Jump up ^ Johnston, Maura (27 March 2012). "The Hives: Go Right Ahead". Rolling Stone. Retrieved 14 July 2014.
Jump up ^ Kent, David (1993). Australian Chart Book 1970–1992. St Ives, N.S.W.: Australian Chart Book. ISBN 0-646-11917-6.
Jump up ^ "Austriancharts.at – Electric Light Orchestra – Don't Bring Me Down" (in German). Ö3 Austria Top 40. Retrieved 1 June 2013.
Jump up ^ "Ultratop.be – Electric Light Orchestra – Don't Bring Me Down" (in Dutch). Ultratop 50. Retrieved 1 June 2013.
Jump up ^ "Top RPM Singles: Issue 6839a." RPM. Library and Archives Canada. Retrieved 1 June 2013.
Jump up ^ "Le Détail par Artiste" (in French). InfoDisc. Select "Electric Light Orchestra" from the artist drop-down menu. Retrieved 15 May 2016.
Jump up ^ "Offiziellecharts.de – Electric Light Orchestra – Don't Bring Me Down". GfK Entertainment Charts. Retrieved 1 June 2013.
Jump up ^ "The Irish Charts – Search Results – Don't Bring Me Down". Irish Singles Chart. Retrieved 6 March 2013.
Jump up ^ "Nederlandse Top 40 – Electric Light Orchestra - Don't Bring Me Down search results" (in Dutch) Dutch Top 40. Retrieved 1 June 2013.
Jump up ^ "Dutchcharts.nl – Electric Light Orchestra – Don't Bring Me Down" (in Dutch). Single Top 100. Retrieved 1 June 2013.
Jump up ^ "Charts.org.nz – Electric Light Orchestra – Don't Bring Me Down". Top 40 Singles. Retrieved 1 June 2013.
Jump up ^ "South African Rock Lists Website SA Charts 1969 – 1989 Acts (E)". Rock.co.za. Retrieved 29 April 2013.
Jump up ^ Salaverri, Fernando (September 2005). Sólo éxitos: año a año, 1959–2002 (in Spanish) (1st ed.). Spain: Fundación Autor-SGAE. ISBN 84-8048-639-2.
Jump up ^ "Swisscharts.com – Electric Light Orchestra – Don't Bring Me Down". Swiss Singles Chart.
Jump up ^ "CASH BOX Top 100 Singles – Week ending SEPTEMBER 22, 1979". Archived from the original on 5 February 2011. . Cash Box.
Jump up ^ "Record World Singles" (PDF). Record World. 15 September 1979. p. 29. ISSN 0034-1622. Retrieved 17 September 2017.
Jump up ^ "Forum – ARIA Charts: Special Occasion Charts – Top 100 End of Year AMR Charts – 1970s". Australian-charts.com. Hung Medien. Retrieved 14 July 2014.
Jump up ^ "Jaaroverzichten 1979" (in Dutch). Ultratop. Hung Medien. Retrieved 14 July 2014.
Jump up ^ "1979 Top 200 Singles". RPM. Vol. 32 no. 13. 22 December 1979. Retrieved 15 May 2016.
Jump up ^ "Top 100-Jaaroverzicht van 1979" (in Dutch). Dutch Top 40. Retrieved 14 July 2014.
Jump up ^ "Jaaroverzichten – Single 1979" (in Dutch). Single Top 100. Hung Medien. Retrieved 14 July 2014.
Jump up ^ "End of Year Charts 1979". Recorded Music New Zealand. Retrieved 30 October 2015.
Jump up ^ "Top 100 Hits for 1979". The Longbored Surfer. Retrieved 14 July 2014.
Jump up ^ "The CASH BOX Year-End Charts: 1979". Archived from the original on 25 August 2012. . Cash Box.
Jump up ^ "British single certifications – ELO – Don't Bring Me Down". British Phonographic Industry. Enter Don't Bring Me Down in the search field and then press Enter.
Jump up ^ "American single certifications – Electric Light Orchestra – Don_t Bring Me Down". Recording Industry Association of America. If necessary, click Advanced, then click Format, then select Single, then click SEARCH
External links[edit]
In-depth Song Analysis at the Jeff Lynne Song Database (jefflynnesongs.com)
Lyrics of this song at MetroLyrics
[hide]
v
t
e
Electric Light Orchestra singles discography
1970s
1972
"10538 Overture" / "First Movement (Jumping Biz)"
"Mr. Radio" (withdrawn)
1973
"Roll Over Beethoven" / "Queen of the Hours"
"Showdown" / "In Old England Town (Instrumental)"
1974
"Ma-Ma-Ma Belle" / "Daybreaker" (US)
"Ma-Ma-Ma Belle" / "Oh No Not Susan" (UK)
"Can't Get It Out of My Head" / "Illusions in G Major"
1975
"Boy Blue" / "Eldorado"
"Evil Woman" / "10538 Overture (live)"
1976
"Strange Magic" / "New World Rising" (US)
"Nightrider" / "Daybreaker (live)"
"Strange Magic" / "Showdown (live)" (UK)
"Livin' Thing" / "Ma-Ma-Ma Belle" (US)
"Livin' Thing" / "Fire On High" (UK)
1977
"Rockaria!" / "Poker"
"Do Ya" / "Nightrider"
"Telephone Line" / "Poor Boy (The Greenwood)" (US)
"Telephone Line" / "Poor Boy (The Greenwood)" and "King of the Universe" (UK)
"Turn to Stone" / "Mister Kingdom"
1978
"Mr. Blue Sky" / "One Summer Dream"
"Sweet Talkin' Woman" / "Fire On High" (US)
"Wild West Hero" / "Eldorado"
"Sweet Talkin' Woman" / "Bluebird is Dead" (UK)
"It's Over" / "The Whale"
The ELO EP
1979
"Shine a Little Love" / "Jungle"
"The Diary of Horace Wimp" / "Down Home Town"
"Don't Bring Me Down" / "Dreaming of 4000"
"Confusion"/"Last Train to London" (UK)
"Confusion" / "Poker" (US)
"Last Train to London" / "Down Home Town" (US)
1980s
1980
"I'm Alive" / "Drum Dreams"
"Xanadu" (duet with Olivia Newton-John) / "Fool Country" (Olivia Newton-John) (UK)
"Xanadu" (with Olivia Newton-John) / "Whenever You're Away from Me" (Olivia Newton-John) (US)
"All Over the World" / "Midnight Blue" (UK)
"All Over the World" / "Drum Dreams" (US)
"Don't Walk Away" / "Across the Border"
1981
"Hold On Tight" / "When Time Stood Still"
"Twilight" / "Julie Don't Live Here"
1982
"Here Is the News"/"Ticket to the Moon"
"Rain Is Falling" / "Another Heart Breaks"
"The Way Life's Meant to Be" / "Wishing"
1983
"Rock 'n' Roll Is King" / "After All"
"Secret Messages" / "Buildings Have Eyes"
"Four Little Diamonds" / "Letter from Spain"
"Stranger" / "Train of Gold"
1986
"Calling America" / "Caught in a Trap"
"So Serious" / "Endless Lies" (US)
"So Serious" / "A Matter of Fact" (UK)
"Getting to the Point" / "Secret Lives"
1990s
1990
"Destination Unknown" (promo)
2000s
2001
"Alright" / "Love Changes All" / "Do Ya" (Unedited Alternative Mix)
"Moment in Paradise" / "Alright" (PBS 2001 Version) / "Mr. Blue Sky" (PBS 2001 Version) / "Livin' Thing" (PBS 2001 Version)
2006
"Surrender"
2007
"Latitude 88 North"
2010s
2015
"When I Was a Boy"
Retrieved from "https://en.wikipedia.org/w/index.php?title=Don%27t_Bring_Me_Down&oldid=808099340"
Categories:
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Geometric distribution
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Geometric
Probability mass function
Cumulative distribution function
Parameters
0 < p < 1 {\displaystyle 0<p<1} success probability (real) 0 < p ≤ 1 {\displaystyle 0<p\leq 1} success probability (real)
Support
k trials where k ∈ { 1 , 2 , 3 , … } {\displaystyle k\in \{1,2,3,\dots \}} k failures where k ∈ { 0 , 1 , 2 , 3 , … } {\displaystyle k\in \{0,1,2,3,\dots \}}
Probability mass function (pmf)
( 1 − p ) k − 1 p {\displaystyle (1-p)^{k-1}p} ( 1 − p ) k p {\displaystyle (1-p)^{k}p}
CDF
1 − ( 1 − p ) k {\displaystyle 1-(1-p)^{k}} 1 − ( 1 − p ) k + 1 {\displaystyle 1-(1-p)^{k+1}}
Mean
1 p {\displaystyle {\frac {1}{p}}} 1 − p p {\displaystyle {\frac {1-p}{p}}}
Median
⌈ − 1 log 2  ( 1 − p ) ⌉ {\displaystyle \left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil }
(not unique if − 1 / log 2  ( 1 − p ) {\displaystyle -1/\log _{2}(1-p)} is an integer)
⌈ − 1 log 2  ( 1 − p ) ⌉ − 1 {\displaystyle \left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil -1}
(not unique if − 1 / log 2  ( 1 − p ) {\displaystyle -1/\log _{2}(1-p)} is an integer)
Mode
1 {\displaystyle 1} 0 {\displaystyle 0}
Variance
1 − p p 2 {\displaystyle {\frac {1-p}{p^{2}}}} 1 − p p 2 {\displaystyle {\frac {1-p}{p^{2}}}}
Skewness
2 − p 1 − p {\displaystyle {\frac {2-p}{\sqrt {1-p}}}} 2 − p 1 − p {\displaystyle {\frac {2-p}{\sqrt {1-p}}}}
Excess kurtosis
6 + p 2 1 − p {\displaystyle 6+{\frac {p^{2}}{1-p}}} 6 + p 2 1 − p {\displaystyle 6+{\frac {p^{2}}{1-p}}}
Entropy
− ( 1 − p ) log 2  ( 1 − p ) − p log 2  p p {\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}} − ( 1 − p ) log 2  ( 1 − p ) − p log 2  p p {\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}}
MGF
p e t 1 − ( 1 − p ) e t , {\displaystyle {\frac {pe^{t}}{1-(1-p)e^{t}}},}
for t < − ln  ( 1 − p ) {\displaystyle t<-\ln(1-p)} p 1 − ( 1 − p ) e t {\displaystyle {\frac {p}{1-(1-p)e^{t}}}}
CF
p e i t 1 − ( 1 − p ) e i t {\displaystyle {\frac {pe^{it}}{1-(1-p)e^{it}}}} p 1 − ( 1 − p ) e i t {\displaystyle {\frac {p}{1-(1-p)e^{it}}}}
In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:
The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}
The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }
Which of these one calls "the" geometric distribution is a matter of convention and convenience.
These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of the number X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.
The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is
Pr ( X = k ) = ( 1 − p ) k − 1 p {\displaystyle \Pr(X=k)=(1-p)^{k-1}p}
for k = 1, 2, 3, ....
The above form of the geometric distribution is used for modeling the number of trials up to and including the first success. By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success:
Pr ( Y = k ) = ( 1 − p ) k p {\displaystyle \Pr(Y=k)=(1-p)^{k}p}
for k = 0, 1, 2, 3, ....
In either case, the sequence of probabilities is a geometric sequence.
For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set { 1, 2, 3, ... } and is a geometric distribution with p = 1/6.
Contents
1 Introduction to the geometric distribution
1.1 Examples
1.2 Assumptions: When is the geometric distribution an appropriate model?
1.3 Probability of outcomes
1.4 Expected number of failures before the first success
2 Moments and cumulants
3 Parameter estimation
4 Other properties
5 Related distributions
6 Computer software for the geometric distribution
6.1 Geometric distribution using R
6.2 Geometric distribution using Excel
7 See also
8 References
9 External links
Introduction to the geometric distribution[edit]
Consider a sequence of trials, where each trial has only two possible outcomes (designated failure and success). The probability of success is assumed to be the same for each trial. In such a sequence of trials, the geometric distribution is useful to model the number of failures before the first success. The distribution gives the probability that there are zero failures before the first success, one failure before the first success, two failures before the first success, and so on.
Examples[edit]
A newlywed couple plans to have children, and will continue until the first girl. What is the probability that there are zero boys before the first girl, one boy before the first girl, two boys before the first girl, and so on?
A doctor is seeking an anti-depressant for a newly diagnosed patient. Suppose that, of the available anti-depressant drugs, the probability that any particular drug will be effective for a particular patient is p=0.6. What is the probability that the first drug found to be effective for this patient is the first drug tried, the second drug tried, and so on? What is the expected number of drugs that will be tried to find one that is effective?
A patient is waiting for a suitable matching kidney donor for a transplant. If the probability that a randomly selected donor is a suitable match is p=0.1, what is the expected number of donors who will be tested before a matching donor is found?
Assumptions: When is the geometric distribution an appropriate model?[edit]
The geometric distribution is an appropriate model if the following assumptions are true.
The phenomenon being modelled is a sequence of independent trials.
There are only two possible outcomes for each trial, often designated success or failure.
The probability of success, p, is the same for every trial.
If these conditions are true, then the geometric random variable is the count of the number of failures before the first success. The possible number of failures before the first success is 0, 1, 2, 3, and so on. The geometric random variable Y is the number of failures before the first success. In the graphs above, this formulation is shown on the right.
An alternative formulation is that the geometric random variable X is the total number of trials up to and including the first success, and the number of failures is X-1. In the graphs above, this formulation is shown on the left.
Probability of outcomes[edit]
Consider the anti-depressant example above. The probability that any given drug is effective (success) is p = 0.6. The probability that a drug will not be effective (fail) is q = 1 – p = 1 – 0.6 = 0.4. Here are probabilities of some possible outcomes.
(i) The first drug works. There are zero failures before the first success. Y = 0 failures. The probability P(zero failures before first success) is simply the probability that the first drug works.
Pr ( Y = 0 ) = q 0 p = 0.4 0 × 0.6 = 1 × 0.6 = 0.6. {\displaystyle \Pr(Y=0)=q^{0}\,p\ =0.4^{0}\times 0.6=1\times 0.6=0.6.}
(ii) The first drug fails, but the second drug works. There is one failure before the first success. Y= 1 failure. The probability for this sequence of events is p(first drug fails) × {\displaystyle \times } p(second drug is success) which is given by
Pr ( Y = 1 ) = q 1 p = 0.4 1 × 0.6 = 0.4 × 0.6 = 0.24. {\displaystyle \Pr(Y=1)=q^{1}\,p\ =0.4^{1}\times 0.6=0.4\times 0.6=0.24.}
(iii) The first drug fails, the second drug fails, but the third drug works. There are two failures before the first success. Y= 2 failures. The probability for this sequence of events is p(first drug fails) × {\displaystyle \times } p(second drug fails) × {\displaystyle \times } p(third drug is success)
Pr ( Y = 2 ) = q 2 p , = 0.4 2 × 0.6 = 0.096. {\displaystyle \Pr(Y=2)=q^{2}\,p,=0.4^{2}\times 0.6=0.096.}
The general formula to calculate the probability of k failures before the first success, where the probability of success is p and the probability of failure is q = 1 − p, is
Pr ( Y = k ) = q k p . {\displaystyle \Pr(Y=k)=q^{k}\,p.}
for k = 0, 1, 2, 3, ....
For the newlyweds awaiting their first girl, the probability of no boys before the first girl is
Pr ( Y = 0 ) = q 0 p = 0.5 0 × 0.5 = 1 × 0.5 = 0.5. {\displaystyle \Pr(Y=0)=q^{0}\,p\ =0.5^{0}\times 0.5=1\times 0.5=0.5.}
The probability of one boy before the first girl is
Pr ( Y = 1 ) = q 1 p = 0.5 1 × 0.5 = 0.5 × 0.5 = 0.25. {\displaystyle \Pr(Y=1)=q^{1}\,p\ =0.5^{1}\times 0.5=0.5\times 0.5=0.25.}
The probability of two boys before the first girl is
Pr ( Y = 2 ) = q 2 p = 0.5 2 × 0.5 = 0.125. {\displaystyle \Pr(Y=2)=q^{2}\,p\ =0.5^{2}\times 0.5=0.125.}
and so on.
Expected number of failures before the first success[edit]
For the geometric distribution, the expected (mean) number of failures before the first success is E(Y) = (1 − p)/p.
For the anti-depressant example, with p = 0.6, the mean number of failures before the first success is E(Y) = (1 − p)/p = (1 − 0.6)/0.6 = 0.67.
For the kidney-donor example, with p = 0.1, the mean number of failures before the first success is E(Y) = (1 − 0.1)/0.1 = 9.
For the alternative formulation, where X is the number of trials up to and including the first success, the expected value is E(X) = 1/p.
Moments and cumulants[edit]
The expected value of a geometrically distributed random variable X is 1/p and the variance is (1 − p)/p2:
E  ( X ) = 1 p , var  ( X ) = 1 − p p 2 . {\displaystyle \operatorname {E} (X)={\frac {1}{p}},\qquad \operatorname {var} (X)={\frac {1-p}{p^{2}}}.}
Similarly, the expected value of the geometrically distributed random variable Y = X − 1 (where Y corresponds to the pmf listed in the right column) is q/p = (1 − p)/p, and its variance is (1 − p)/p2:
E  ( Y ) = 1 − p p , var  ( Y ) = 1 − p p 2 . {\displaystyle \operatorname {E} (Y)={\frac {1-p}{p}},\qquad \operatorname {var} (Y)={\frac {1-p}{p^{2}}}.}
Let μ = (1 − p)/p be the expected value of Y. Then the cumulants κ n {\displaystyle \kappa _{n}} of the probability distribution of Y satisfy the recursion
κ n + 1 = μ ( μ + 1 ) d κ n d μ . {\displaystyle \kappa _{n+1}=\mu (\mu +1){\frac {d\kappa _{n}}{d\mu }}.}
Outline of proof: That the expected value is (1 − p)/p can be shown in the following way. Let Y be as above. Then
E ( Y ) = ∑ k = 0 ∞ ( 1 − p ) k p ⋅ k = p ∑ k = 0 ∞ ( 1 − p ) k k = p ( 1 − p ) ∑ k = 0 ∞ ( 1 − p ) k − 1 ⋅ k = p ( 1 − p ) [ d d p ( − ∑ k = 0 ∞ ( 1 − p ) k ) ] = p ( 1 − p ) d d p ( − 1 p ) = 1 − p p . {\displaystyle {\begin{aligned}\mathrm {E} (Y)&{}=\sum _{k=0}^{\infty }(1-p)^{k}p\cdot k\\&{}=p\sum _{k=0}^{\infty }(1-p)^{k}k\\&{}=p(1-p)\sum _{k=0}^{\infty }(1-p)^{k-1}\cdot k\\&{}=p(1-p)\left[{\frac {d}{dp}}\left(-\sum _{k=0}^{\infty }(1-p)^{k}\right)\right]\\&{}=p(1-p){\frac {d}{dp}}\left(-{\frac {1}{p}}\right)={\frac {1-p}{p}}.\end{aligned}}}
(The interchange of summation and differentiation is justified by the fact that convergent power series converge uniformly on compact subsets of the set of points where they converge.)
Parameter estimation[edit]
For both variants of the geometric distribution, the parameter p can be estimated by equating the expected value with the sample mean. This is the method of moments, which in this case happens to yield maximum likelihood estimates of p.[citation needed]
Specifically, for the first variant let k = k1, ..., kn be a sample where ki ≥ 1 for i = 1, ..., n. Then p can be estimated as
p ^ = ( 1 n ∑ i = 1 n k i ) − 1 = n ∑ i = 1 n k i . {\displaystyle {\widehat {p}}=\left({\frac {1}{n}}\sum _{i=1}^{n}k_{i}\right)^{-1}={\frac {n}{\sum _{i=1}^{n}k_{i}}}.\!}
In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta(α, β) prior, then the posterior distribution is[citation needed]
p ∼ B e t a ( α + n , β + ∑ i = 1 n ( k i − 1 ) ) . {\displaystyle p\sim \mathrm {Beta} \left(\alpha +n,\ \beta +\sum _{i=1}^{n}(k_{i}-1)\right).\!}
The posterior mean E[p] approaches the maximum likelihood estimate p ^ {\displaystyle {\widehat {p}}} as α and β approach zero.
In the alternative case, let k1, ..., kn be a sample where ki ≥ 0 for i = 1, ..., n. Then p can be estimated as
p ^ = ( 1 + 1 n ∑ i = 1 n k i ) − 1 = n ∑ i = 1 n k i + n . {\displaystyle {\widehat {p}}=\left(1+{\frac {1}{n}}\sum _{i=1}^{n}k_{i}\right)^{-1}={\frac {n}{\sum _{i=1}^{n}k_{i}+n}}.\!}
The posterior distribution of p given a Beta(α, β) prior is[citation needed]
p ∼ B e t a ( α + n , β + ∑ i = 1 n k i ) . {\displaystyle p\sim \mathrm {Beta} \left(\alpha +n,\ \beta +\sum _{i=1}^{n}k_{i}\right).\!}
Again the posterior mean E[p] approaches the maximum likelihood estimate p ^ {\displaystyle {\widehat {p}}} as α and β approach zero.
Other properties[edit]
The probability-generating functions of X and Y are, respectively,
G X ( s ) = s p 1 − s ( 1 − p ) , G Y ( s ) = p 1 − s ( 1 − p ) , | s | < ( 1 − p ) − 1 . {\displaystyle {\begin{aligned}G_{X}(s)&={\frac {s\,p}{1-s\,(1-p)}},\\[10pt]G_{Y}(s)&={\frac {p}{1-s\,(1-p)}},\quad |s|<(1-p)^{-1}.\end{aligned}}}
Like its continuous analogue (the exponential distribution), the geometric distribution is memoryless. That means that if you intend to repeat an experiment until the first success, then, given that the first success has not yet occurred, the conditional probability distribution of the number of additional trials does not depend on how many failures have been observed. The die one throws or the coin one tosses does not have a "memory" of these failures. The geometric distribution is the only memoryless discrete distribution.
Among all discrete probability distributions supported on {1, 2, 3, ... } with given expected value μ, the geometric distribution X with parameter p = 1/μ is the one with the largest entropy.[citation needed]
The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y1, ..., Yn whose sum has the same distribution that Y has. These will not be geometrically distributed unless n = 1; they follow a negative binomial distribution.
The decimal digits of the geometrically distributed random variable Y are a sequence of independent (and not identically distributed) random variables.[citation needed] For example, the hundreds digit D has this probability distribution:
Pr ( D = d ) = q 100 d 1 + q 100 + q 200 + ⋯ + q 900 , {\displaystyle \Pr(D=d)={q^{100d} \over 1+q^{100}+q^{200}+\cdots +q^{900}},}
where q = 1 − p, and similarly for the other digits, and, more generally, similarly for numeral systems with other bases than 10. When the base is 2, this shows that a geometrically distributed random variable can be written as a sum of independent random variables whose probability distributions are indecomposable.
Golomb coding is the optimal prefix code[clarification needed] for the geometric discrete distribution.[citation needed]
Related distributions[edit]
The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y1, ..., Yr are independent geometrically distributed variables with parameter p, then the sum
Z = ∑ m = 1 r Y m {\displaystyle Z=\sum _{m=1}^{r}Y_{m}}
follows a negative binomial distribution with parameters r and p.[1]
The geometric distribution is a special case of discrete compound Poisson distribution.
If Y1, ..., Yr are independent geometrically distributed variables (with possibly different success parameters pm), then their minimum
W = min m ∈ 1 , … , r Y m {\displaystyle W=\min _{m\in 1,\ldots ,r}Y_{m}\,}
is also geometrically distributed, with parameter p = 1 − ∏ m ( 1 − p m ) . {\displaystyle p=1-\prod _{m}(1-p_{m}).} [citation needed]
Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable Xk has a Poisson distribution with expected value r k/k. Then
∑ k = 1 ∞ k X k {\displaystyle \sum _{k=1}^{\infty }k\,X_{k}}
has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value r/(1 − r).[citation needed]
The exponential distribution is the continuous analogue of the geometric distribution. If X is an exponentially distributed random variable with parameter λ, then
Y = ⌊ X ⌋ , {\displaystyle Y=\lfloor X\rfloor ,}
where ⌊ ⌋ {\displaystyle \lfloor \quad \rfloor } is the floor (or greatest integer) function, is a geometrically distributed random variable with parameter p = 1 − e−λ (thus λ = −ln(1 − p)[2]) and taking values in the set {0, 1, 2, ...}. This can be used to generate geometrically distributed pseudorandom numbers by first generating exponentially distributed pseudorandom numbers from a uniform pseudorandom number generator: then ⌊ ln  ( U ) / ln  ( 1 − p ) ⌋ {\displaystyle \lfloor \ln(U)/\ln(1-p)\rfloor } is geometrically distributed with parameter p {\displaystyle p} , if U {\displaystyle U} is uniformly distributed in [0,1].
If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since
P ( X > a ) = ( 1 − p ) a = ( 1 − 1 n ) n 1 n ( a ) = [ ( 1 − 1 n ) n ] 1 n ( a ) → [ e − 1 ] 1 n ( a ) = e − 1 n a as n → ∞ . {\displaystyle {\begin{aligned}P(X>a)&=(1-p)^{a}=\left(1-{\frac {1}{n}}\right)^{n{\frac {1}{n}}(a)}=\left[\left(1-{\frac {1}{n}}\right)^{n}\right]^{{\frac {1}{n}}(a)}\\&\to [e^{-1}]^{{\frac {1}{n}}(a)}=e^{-{\frac {1}{n}}a}{\text{ as }}n\to \infty .\end{aligned}}}
Computer software for the geometric distribution[edit]
Geometric distribution using R[edit]
The R function dgeom(k, prob) calculates the probability that there are k failures before the first success, where the argument "prob" is the probability of success on each trial.
For example,
dgeom(0,0.6) = 0.6
dgeom(1,0.6) = 0.24
R uses the convention that k is the number of failures, so that the number of trials up to and including the first success is k + 1.
The following R code creates a graph of the geometric distribution from Y = 0 to 10, with p = 0.6.
Y=0:10
plot(Y, dgeom(Y,0.6), type="h", ylim=c(0,1), main="Geometric distribution for p=0.6", ylab="P(Y=Y)", xlab="Y=Number of failures before first success")
Geometric distribution using Excel[edit]
The geometric distribution, for the number of failures before the first success, is a special case of the negative binomial distribution, for the number of failures before s successes.
The Excel function NEGBINOMDIST(number_f, number_s, probability_s) calculates the probability of k = number_f failures before s = number_s successes where p = probability_s is the probability of success on each trial. For the geometric distribution, let number_s = 1 success.
For example,
=NEGBINOMDIST(0, 1, 0.6) = 0.6
=NEGBINOMDIST(1, 1, 0.6) = 0.24
Like R, Excel uses the convention that k is the number of failures, so that the number of trials up to and including the first success is k + 1.
See also[edit]
Hypergeometric distribution
Coupon collector's problem
Compound Poisson distribution
Negative binomial distribution
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (March 2011) (Learn how and when to remove this template message)
References[edit]
Jump up ^ Pitman, Jim. Probability (1993 edition). Springer Publishers. pp 372.
Jump up ^ "Wolfram-Alpha: Computational Knowledge Engine". www.wolframalpha.com.
External links[edit]
"Geometric distribution". PlanetMath.
Geometric distribution on MathWorld.
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Radical Whigs
From Wikipedia, the free encyclopedia
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The Radical Whigs were a group of British political commentators associated with the British Whig faction who were at the forefront of the Radical movement.
Contents
[hide]
1 Seventeenth century
2 Eighteenth century
3 References
4 See also
Seventeenth century[edit]
The radical Whigs ideology "arose from a series of political upheavals in seventeenth-century England: the English Civil War, the exclusion crisis of 1679-81, and the Glorious Revolution of 1688. Broadly speaking, this Whig theory described two sorts of threats to political freedom: a general moral decay which would invite the intrusion of evil and despotic rulers, and the encroachment of executive authority upon the legislature, the attempt that power always made to subdue the liberty protected by mixed government."[1] This political theory was mainly based on the writings of John Milton, John Locke, James Harrington, and Algernon Sydney.[2]
Eighteenth century[edit]
The eighteenth-century Whigs, or commonwealthmen, in particular John Trenchard, Thomas Gordon, and Benjamin Hoadly, "praised the mixed constitution of monarchy, aristocracy, and democracy, and they attributed English liberty to it; and like Locke they postulated a state of nature from which rights arose which the civil polity, created by mutual consent, guaranteed; they argued that a contract formed government and that sovereignty resided in the people."[3]
The radical Whigs' political ideas played a significant role in the development of the American Revolution, as their republican writings were widely read by the American colonists, many of whom were convinced by their reading that they should be very watchful for any threats to their liberties. Subsequently, when the colonists were indignant about their perceived lack of representation and taxes such as the Stamp Act, the Sugar Act and the Tea Act, the colonists broke away from the Kingdom of Great Britain to form the United States of America. "Radical Whig perceptions of politics attracted widespread support in America because they revived the traditional concerns of a Protestant culture that had always verged on Puritanism. That moral decay threatened free government could not come as a surprise to a people whose fathers had fled England to escape sin. The importance of virtue, frugality, industry, and calling was at the heart of their moral code. An overbearing executive, and the threat of corruption through idle, useless officials, or placemen, had figured prominently in their explanations of their exile in America." So "most" of the ideas the American Revolutionaries put into their political system "were a part of the great tradition of the eighteenth-century commonwealthmen."[4][5]
Although the commonwealthmen had little influence on British politics in the eighteenth century, their political ideas had a long-term effect on Britain's constitutional system: constitutional monarchy, which in turn became a model for other countries, for example, Sweden, Norway, Denmark, the Netherlands, and Belgium.[6]
References[edit]
Jump up ^ Robert Middlekauff (2005), The Glorious Cause: The American Revolution, 1763-1789, Revised and Expanded Edition, Oxford University Press, ISBN 978-0-19-516247-9, p. 51
Jump up ^ Middlekauff (2005), p. 136
Jump up ^ Middlekauff (2005 ), p. 136
Jump up ^ Middlekauff (2005), pp. 51-52
Jump up ^ Cf. Thomas S. Kidd (2010), God of Liberty: A Religious History of the American Revolution, New York, N.Y., ISBN 978-0-465-00235-1, pp. 7-8
Jump up ^ Heinrich August Winkler (2012), Geschichte des Westens. Von den Anfängen in der Antike bis zum 20. Jahrhundert, Second, Revised Edition, Munich (Germany), ISBN 978 3 406 59235 5, pp. 142ff, 634ff
See also[edit]
Levellers movement
Patriot (American Revolution)
Patriot Whigs
Philosophic Whigs
Political radicalism
Radical movement
Whig (disambiguation)

This article relating to the history of the United States is a stub. You can help Wikipedia by expanding it.
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Retrieved from "https://en.wikipedia.org/w/index.php?title=Radical_Whigs&oldid=791889916"
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