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| {{Infobox probability distribution 2
| | == 諺にトラブルの根を行くよう == |
| | name = Geometric
| |
| | type = mass
| |
| | pdf_image = [[File:geometric pmf.svg|450px]]
| |
| | cdf_image = [[File:geometric cdf.svg|450px]]
| |
| | parameters = <math>0< p \leq 1</math> success probability ([[real number|real]])
| |
| | support = <math>k \in \{1,2,3,\dots\}\!</math>
| |
| | pdf = <math>(1 - p)^{k-1}\,p\!</math>
| |
| | cdf = <math>1-(1 - p)^k\!</math>
| |
| | mean = <math>\frac{1}{p}\!</math>
| |
| | median = <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil\!</math> (not unique if <math>-1/\log_2(1-p)</math> is an integer)
| |
| | mode = <math>1</math>
| |
| | variance = <math>\frac{1-p}{p^2}\!</math>
| |
| | skewness = <math>\frac{2-p}{\sqrt{1-p}}\!</math>
| |
| | kurtosis = <math>6+\frac{p^2}{1-p}\!</math>
| |
| | entropy = <math>\tfrac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}\!</math>
| |
| | mgf = <math>\frac{pe^t}{1-(1-p) e^t}\!</math>, <br> <big> <big>for</big></big> <math>t<-\ln(1-p)\!</math>
| |
| | char = <math>\frac{pe^{it}}{1-(1-p)\,e^{it}}\!</math>
| |
| | parameters2 = <math>0< p \leq 1</math> success probability ([[real number|real]])
| |
| | support2 = <math>k \in \{0,1,2,3,\dots\}\!</math>
| |
| | pdf2 = <math>(1 - p)^{k}\,p\!</math>
| |
| | cdf2 = <math>1-(1 - p)^{k+1}\!</math>
| |
| | mean2 = <math>\frac{1-p}{p}\!</math>
| |
| | median2 = <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil\! - 1</math> (not unique if <math>-1/\log_2(1-p)</math> is an integer)
| |
| | mode2 = <math>0</math>
| |
| | variance2 = <math>\frac{1-p}{p^2}\!</math>
| |
| | skewness2 = <math>\frac{2-p}{\sqrt{1-p}}\!</math>
| |
| | kurtosis2 = <math>6+\frac{p^2}{1-p}\!</math>
| |
| | entropy2 = <math>\tfrac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}\!</math>
| |
| | mgf2 = <math>\frac{p}{1-(1-p)e^t}\!</math>
| |
| | char2 = <math>\frac{p}{1-(1-p)\,e^{it}}\!</math>
| |
| }}
| |
| In [[probability theory]] and [[statistics]], the '''geometric distribution''' is either of two [[discrete probability distribution]]s:
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| * The probability distribution of the number ''X'' of [[Bernoulli trial]]s needed to get one success, supported on the set { 1, 2, 3, ...}
| | 「DOまたは薫の子供のためにため？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] '<br>諺にトラブルの根を行くよう<br>シャオヤンの心の笑顔が、発言はしない悲惨な薫子供やレベルの種類はもちろんのこと、ありません、彼は古代の職業を並べる予感を持って、私は怖いですし、滑らかではありません<br>ゆっくりと街に歩いた小型医療セントで<br>が、シャオヤンのコマンド内の5以来の躊躇後者、バオ泉によると、過去歩いた： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 腕時計 ソーラー 電波] '控えめな場合は、彼は時間を見つけることができるようになります、とシャオヤンあなたが学ぶ？私はあなたが本当にミスを伴うことができるかどうかを確認するために興味があった！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 時計 価格] '<br><br>最後の文、薄い蚊が、それは非常に明確にシャオヤンの耳、中に入るとシャオヤンの言葉を聞いている、また、本当にため息をため息<br>「ああ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド]。」<br><br><br>シャオヤンはゆっくりと言い訳を探していなかった、彼の頭をうなずいて、彼は明らかに彼女の気質の才能に古代の部族の間で子のステータスを吸っ |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://www.kanko-hen.com/cgi-bin/buccgi-ruby-bbs/kanko-hen-bbs.cgi http://www.kanko-hen.com/cgi-bin/buccgi-ruby-bbs/kanko-hen-bbs.cgi]</li> |
| | |
| | <li>[http://a-g.ru/forum/viewtopic.php?f=2&t=157461 http://a-g.ru/forum/viewtopic.php?f=2&t=157461]</li> |
| | |
| | <li>[http://cosplaythai.com/index.php?p=blogs/viewstory/139874 http://cosplaythai.com/index.php?p=blogs/viewstory/139874]</li> |
| | |
| | </ul> |
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| * The probability distribution of the number ''Y'' = ''X'' − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... }
| | == は、現在光沢「ファン」醸し出しています == |
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| Which of these one calls "the" geometric distribution is a matter of convention and convenience.
| | 次に、すぐに、お香のようなダンエッセンスのように、急に充填され、オープン [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 掛け時計]<br>「ハハ、私はそれを作った！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン] '<br><br>彼の手の中に少しバイアスされた白人男性、彼の背中笑い、ダン「医学」の竜眼の大きさに作られた目の下の点で<br>は、現在光沢「ファン」醸し出しています [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 腕時計 電波 ソーラー]。<br>男性のための<br>この失態の笑いに関係なく、誰も被告人の存在に触れていないが、前者の目に見てみると、精錬精錬 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 説明書] '医学'教祖ダン7の中間財 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock] '医学'が可能ないくつかのホット、いずれかを約束しているどこに絶対的なVIP待遇である。<br>ダンはすでに後半に入力されます、そのような時期に、錬金術を続け、二人だけ<br>は今、人は本当に別の偉大な能力はもちろん、ある一人一人を持って、この状況は当然、すべての後に、かつての大多数ではない神経の多くの人ができ恥のようなもの、この場合のポイント |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://bbs.28jk.com/forum.php?mod=viewthread&tid=738792 http://bbs.28jk.com/forum.php?mod=viewthread&tid=738792]</li> |
| | |
| | <li>[http://www.software77.com/cgi-bin/guestbook/guestbook.cgi http://www.software77.com/cgi-bin/guestbook/guestbook.cgi]</li> |
| | |
| | <li>[http://www.burgkultur.at/forum/guestbook/guestbook.cgi http://www.burgkultur.at/forum/guestbook/guestbook.cgi]</li> |
| | |
| | </ul> |
|
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| 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.
| | == 13399 == |
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| It’s the probability that the first occurrence of success require k number of independent trials, each with success probability p. If the probability of success on each trial is ''p'', then the probability that the ''k''th trial (out of ''k'' trials) is the first success is
| | 罪は微笑んで彼のあご、Lianbuqingyiをうなずいた、フォローアップ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ]。<br><br>葉のフロントヤードは、ここでは現時点では、ここに囲まれたシルエットが多数、雰囲気は特に緊張している [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書]。<br><br>ゆったり前庭には、自然がイェ家族や他の人であるが、もう一方の側が、ほぼ道路はピンクのローブシルエットに身を包んで、明確に異なる2つの陣営に分離された、これらの人々はまっすぐ立って、運動量がかなりあることが今曹操ダン地域の位置と、によって、家族や他の人だけでなく、もちろん優越感のようなものの目に見える横暴、反対の葉は確かに、家を比較することはできないままになります [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 激安]。<br><br>曹操、最前線で歩行者は、2赤いローブ老人、二人の男と無関心の外観、壮大な勢いで満たされた全身で、彼の手には、「挿入」それは、もちろん、人を見て珍しいことではありませんようにスリーブとの間に、それは、ほとんど見え人々が気に持っているので、アイデアは、彼らが持っていることではなく、高齢者の灰色のローブの前で二人の男 |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://117.21.174.74/upload/forum.php?mod=viewthread&tid=3219673&fromuid=380545 http://117.21.174.74/upload/forum.php?mod=viewthread&tid=3219673&fromuid=380545]</li> |
| | |
| | <li>[http://fuzdao.com/forum.php?mod=viewthread&tid=17526&fromuid=4366 http://fuzdao.com/forum.php?mod=viewthread&tid=17526&fromuid=4366]</li> |
| | |
| | <li>[http://www.ploiesti.ro/cgi-bin/guestbook/guestbook.cgi http://www.ploiesti.ro/cgi-bin/guestbook/guestbook.cgi]</li> |
| | |
| | </ul> |
|
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| :<math>\Pr(X = k) = (1-p)^{k-1}\,p\,</math>
| | == シャオヤン輝いていたの目に == |
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| for ''k'' = 1, 2, 3, ....
| | また、あなたがそれを行うことができれば、私は小岩この人生、そしてあなたに、一つのことをお願いし、「今日：経験は、そのように、あまりにも多く、ここでストレートメデューサを見つめ、目滞留せず、ささやい害はありません。 [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml] '<br>これは懇願の言葉でそう言って、彼女の前で彼の最初の時間ですので、奇妙な魅力メドゥーサ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン] '混乱'で埋め<br>聞いたが、この瞬間に、理解するので、やや狭め彼女をシャオヤンの狭い目を強制、後者の骨誇りに漠然と感情の種類を気に 'セックス'と、彼女は非常に明確、しかし、今日です<br>念頭に置いて<br>奇妙ゆっくり抑うつ気分は、女王メデューサの光は遠くない昔からの「医学」に懸濁時計本体を見て、法執行の前にアヒルチラリ、言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオの時計] '私はそのショットがあなたを守りたい先生？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波] '<br>「ああ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 腕時計 メンズ casio]。」<br><br>シャオヤン輝いていたの目に<br>、それは良い瞬間の後、あまりにも遠く、遠くない視線Meimouアヒルの法執行機関からゆっくりと部分的なメデューサです |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>[http://www.aiao168.com/forum.php?mod=viewthread&tid=248919&fromuid=128011 http://www.aiao168.com/forum.php?mod=viewthread&tid=248919&fromuid=128011]</li> |
| | |
| | <li>[http://tm1.math.uh.edu/search.cgi http://tm1.math.uh.edu/search.cgi]</li> |
| | |
| | <li>[http://www.thecastle-pub.com/cgi-bin/guestbook/guestbook.cgi http://www.thecastle-pub.com/cgi-bin/guestbook/guestbook.cgi]</li> |
| | |
| | </ul> |
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| The above form of geometric distribution is used for modeling the number of trials until the first success. By contrast, the following form of geometric distribution is used for modeling number of failures until the first success:
| | == '彼と誰もメインtightlippedロード、シャオヤンの動きを見た == |
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| :<math>\Pr(Y=k) = (1 - p)^k\,p\,</math> | | 主の口が悲観的な笑顔を誘発する、ゆっくりと言った [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計]。<br>トレーニングフィールドにわたって掃引<br>シャオヤンの目、この流血の傭兵グループのメンバーに加えてその家族、少なくとも四から五〇〇の人は、この変更1を持っている、それは彼が4〜500「火の蓮のボトルを考え出す必要があったと言うことです「うわあ、この古い男の食欲は、本当に素晴らしい<br><br>「どうやって？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] '彼と誰もメインtightlippedロード、シャオヤンの動きを見た。<br><br>「非常に良い」シャオヤンは、微笑んで言葉を吐き出すが、それはそれらの血まみれの傭兵グループが、また顔」カラー「グレー、期待の心ながら、インスタント冷たい彼は主表面 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] '色'だったすべての家族を作ることですゆっくり打ち砕か。<br>'ケースであること、私たちは妻はあなたが珍しいことではないことを知っているが、あなたは本当に正しい、私の家族と彼をしたい場合、私は彼が家に、それは、ノーバディではない、私はあなたが多くの利点を持っていない恐れている [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 電波時計]。にもかかわらず、残すためにあなたを要求する「<br>ではない、彼はいくつかのためらい上にある場合、彼は、すべての家族主を冷笑 |
| | | 相关的主题文章： |
| for ''k'' = 0, 1, 2, 3, ....
| | <ul> |
| | | |
| In either case, the sequence of probabilities is a [[geometric sequence]].
| | <li>[http://www.uuapen.com/home.php?mod=space&uid=920047 http://www.uuapen.com/home.php?mod=space&uid=920047]</li> |
| | | |
| For example, suppose an ordinary [[dice|die]] <!-- "die" is the correct singular form of the plural "dice." --> 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.
| | <li>[http://www.taoshouyou.com/home.php?mod=space&uid=103794 http://www.taoshouyou.com/home.php?mod=space&uid=103794]</li> |
| | | |
| ==Moments and cumulants==
| | <li>[http://rj.chinayanfang.cn/bbs/home.php?mod=space&uid=68770 http://rj.chinayanfang.cn/bbs/home.php?mod=space&uid=68770]</li> |
| | | |
| The [[expected value]] of a geometrically distributed [[random variable]] ''X'' is 1/''p'' and the [[variance]] is (1 − ''p'')/''p''<sup>2</sup>:
| | </ul> |
| | |
| :<math>\mathrm{E}(X) = \frac{1}{p},
| |
| \qquad\mathrm{var}(X) = \frac{1-p}{p^2}.</math>
| |
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| Similarly, the expected value of the geometrically distributed random variable ''Y'' (where Y corresponds to the pmf listed in the right column) is (1 − ''p'')/''p'', and its variance is (1 − ''p'')/''p''<sup>2</sup>:
| |
| | |
| :<math>\mathrm{E}(Y) = \frac{1-p}{p},
| |
| \qquad\mathrm{var}(Y) = \frac{1-p}{p^2}.</math>
| |
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| Let ''μ'' = (1 − ''p'')/''p'' be the expected value of ''Y''. Then the [[cumulant]]s <math>\kappa_n</math> of the probability distribution of ''Y'' satisfy the recursion
| |
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| :<math>\kappa_{n+1} = \mu(\mu+1) \frac{d\kappa_n}{d\mu}.</math>
| |
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| ''Outline of proof:'' That the expected value is (1 − ''p'')/''p'' can be shown in the following way. Let ''Y'' be as above. Then
| |
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| : <math>
| |
| \begin{align}
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| \mathrm{E}(Y) & {} =\sum_{k=0}^\infty (1-p)^k p\cdot k \\
| |
| & {} =p\sum_{k=0}^\infty(1-p)^k k \\
| |
| & {} = p\left[\frac{d}{dp}\left(-\sum_{k=0}^\infty (1-p)^k\right)\right](1-p) \\
| |
| & {} =-p(1-p)\frac{d}{dp}\frac{1}{p}=\frac{1-p}{p}.
| |
| \end{align}
| |
| </math>
| |
| | |
| (The interchange of summation and differentiation is justified by the fact that convergent [[power series]] [[uniform convergence|converge uniformly]] on [[compact space|compact]] subsets of the set of points where they converge.)
| |
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| ==Parameter estimation==
| |
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| 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 (statistics)|method of moments]], which in this case happens to yield [[maximum likelihood]] estimates of ''p''.{{cn|date=May 2012}}
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| Specifically, for the first variant let ''k'' = ''k''<sub>1</sub>, ..., ''k''<sub>''n''</sub> be a [[sample (statistics)|sample]] where ''k''<sub>''i''</sub> ≥ 1 for ''i'' = 1, ..., ''n''. Then ''p'' can be estimated as
| |
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| :<math>\widehat{p} = \left(\frac1n \sum_{i=1}^n k_i\right)^{-1} = \frac{n}{\sum_{i=1}^n k_i }. \!</math>
| |
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| 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 distribution|prior]], then the [[posterior distribution]] is{{cn|date=May 2012}}
| |
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| :<math>p \sim \mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^n (k_i-1)\right). \!</math>
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| The posterior mean E[''p''] approaches the maximum likelihood estimate <math>\widehat{p}</math> as ''α'' and ''β'' approach zero.
| |
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| In the alternative case, let ''k''<sub>1</sub>, ..., ''k''<sub>''n''</sub> be a sample where ''k''<sub>''i''</sub> ≥ 0 for ''i'' = 1, ..., ''n''. Then ''p'' can be estimated as
| |
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| :<math>\widehat{p} = \left(1 + \frac1n \sum_{i=1}^n k_i\right)^{-1}. \!</math>
| |
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| The posterior distribution of ''p'' given a Beta(''α'', ''β'') prior is{{cn|date=May 2012}}
| |
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| :<math>p \sim \mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^n k_i\right). \!</math>
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| Again the posterior mean E[''p''] approaches the maximum likelihood estimate <math>\widehat{p}</math> as ''α'' and ''β'' approach zero.
| |
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| == Other Properties ==
| |
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| * The [[probability-generating function]]s of ''X'' and ''Y'' are, respectively,
| |
| ::<math>
| |
| \begin{align}
| |
| 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{align}
| |
| </math>
| |
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| * Like its continuous analogue (the [[exponential distribution]]), the geometric distribution is [[memorylessness|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.
| |
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| * 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 [[maximum entropy probability distribution|largest entropy]].{{cn|date=May 2012}}
| |
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| * The geometric distribution of the number ''Y'' of failures before the first success is [[infinite divisibility (probability)|infinitely divisible]], i.e., for any positive integer ''n'', there exist independent identically distributed random variables ''Y''<sub>1</sub>, ..., ''Y''<sub>''n''</sub> whose sum has the same distribution that ''Y'' has. These will not be geometrically distributed unless ''n'' = 1; they follow a [[negative binomial distribution]].
| |
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| * The decimal digits of the geometrically distributed random variable ''Y'' are a sequence of [[statistical independence|independent]] (and ''not'' identically distributed) random variables.{{cn|date=May 2012}} For example, the <!-- "hundreds" is correct; "hundredth" is wrong -->hundreds<!-- "hundreds" is correct; "hundredth" is wrong --> digit ''D'' has this probability distribution:
| |
| ::<math>\Pr(D=d) = {q^{100d} \over 1 + q^{100} + q^{200} + \cdots + q^{900}},</math>
| |
| :where ''q'' = 1 − ''p'', and similarly for the other digits, and, more generally, similarly for [[numeral system]]s 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 distribution|indecomposable]].
| |
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| * [[Golomb coding]] is the optimal [[prefix code]]{{clarify|date=May 2012}} for the geometric discrete distribution.{{cn|date=May 2012}}
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| ==Related distributions==
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| * The geometric distribution ''Y'' is a special case of the [[negative binomial distribution]], with ''r'' = 1. More generally, if ''Y''<sub>1</sub>, ..., ''Y''<sub>''r''</sub> are [[statistical independence|independent]] geometrically distributed variables with parameter ''p'', then the sum
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| ::<math>Z = \sum_{m=1}^r Y_m</math>
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| :follows a negative binomial distribution with parameters ''r'' and ''p''.<ref>Pitman, Jim. Probability (1993 edition). Springer Publishers. pp 372.</ref> | |
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| * If ''Y''<sub>1</sub>, ..., ''Y''<sub>''r''</sub> are independent geometrically distributed variables (with possibly different success parameters ''p''<sub>''m''</sub>), then their [[minimum]]
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| ::<math>W = \min_{m \in 1, \dots, r} Y_m\,</math>
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| :is also geometrically distributed, with parameter <math>p = 1-\prod_m(1-p_{m}).</math>{{cn|date=May 2012}}
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| * Suppose 0 < ''r'' < 1, and for ''k'' = 1, 2, 3, ... the random variable ''X''<sub>''k''</sub> has a [[Poisson distribution]] with expected value ''r''<sup> ''k''</sup>/''k''. Then
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| ::<math>\sum_{k=1}^\infty k\,X_k</math>
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| :has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value ''r''/(1 − ''r'').{{cn|date=May 2012}}
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| * The [[exponential distribution]] is the continuous analogue of the geometric distribution. If ''X'' is an exponentially distributed random variable with parameter λ, then
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| ::<math>Y = \lfloor X \rfloor,</math>
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| : where <math>\lfloor \quad \rfloor</math> is the [[Floor and ceiling functions|floor]] (or greatest integer) function, is a geometrically distributed random variable with parameter ''p'' = 1 − ''e''<sup>−''λ''</sup> (thus ''λ'' = −ln(1 − ''p'')<ref>http://www.wolframalpha.com/input/?i=inverse+p+%3D+1+-+e^-l</ref>) and taking values in the set {0, 1, 2, ...}. This can be used to generate geometrically distributed pseudorandom numbers by first [[Exponential distribution#Generating exponential variates|generating exponentially distributed]] pseudorandom numbers from a uniform [[pseudorandom number generator]]: then <math>\lfloor \ln(U) / \ln(1-p)\rfloor</math> is geometrically distributed with parameter <math>p</math>, if <math>U</math> is uniformly distributed in [0,1].
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| == See also == | |
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| * [[Hypergeometric distribution]]
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| * [[Coupon collector's problem]]
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| {{More footnotes|date=March 2011}}
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{planetmath reference|id=3456|title=Geometric distribution}}
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| *[http://mathworld.wolfram.com/GeometricDistribution.html Geometric distribution] on [[MathWorld]].
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| *[http://www.solvemymath.com/online_math_calculator/statistics/discrete_distributions/geometric/index.php Online geometric distribution calculator]
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| * [http://www.elektro-energetika.cz/calculations/distrgeo.php?language=english Online calculator of Geometric distribution]
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| {{ProbDistributions|discrete-infinite}}
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| {{Common univariate probability distributions}}
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| [[Category:Discrete distributions]]
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| [[Category:Exponential family distributions]]
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| [[Category:Infinitely divisible probability distributions]]
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| [[Category:Probability distributions]]
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