From Wikipedia, the free encyclopedia
Side by side comparison{{Infobox probability distribution}} | {{Infobox probability distribution/sandbox}} |
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Binomial distribution
Probability mass function ![Probability mass function for the binomial distribution](//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Binomial_distribution_pmf.svg/300px-Binomial_distribution_pmf.svg.png) |
Cumulative distribution function ![Cumulative distribution function for the binomial distribution](//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/300px-Binomial_distribution_cdf.svg.png) | Notation |
![{\displaystyle B(n,p)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/58280f6b0f1a1b474a7047c07943f908e775aa71) |
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Parameters |
– number of trials
– success probability for each trial |
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Support |
– number of successes |
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PMF |
![{\displaystyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e8a14d7d4b8ca69d8164da826f6c0c683254) |
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CDF |
![{\displaystyle I_{1-p}(n-k,1+k)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5e0094516ca80b89715e5ab3c00d033b1958389c) |
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Mean |
![{\displaystyle np}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27) |
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Median |
or ![{\displaystyle \lceil np\rceil }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e7ade49121f3f129664d81676c95c51302a43653) |
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Mode |
or ![{\displaystyle \lceil (n+1)p\rceil -1}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a054d92b493838f6bfe891a53e1629bdf8f85fb4) |
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Variance |
![{\displaystyle np(1-p)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/57f093250a1d822df677a03ac8aa78c6a8029866) |
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Skewness |
![{\displaystyle {\frac {1-2p}{\sqrt {np(1-p)}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a43fb738bf588b4cd915d98195b64b5c274dfe61) |
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Excess kurtosis |
![{\displaystyle {\frac {1-6p(1-p)}{np(1-p)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a4c21cc7952fe9e6fbe001e2339d9f3d021846d3) |
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Entropy |
![{\displaystyle {\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/51f7c6331e9bb5bc6f03fc5e34f795d4b025ea89) in shannons. For nats, use the natural log in the log. |
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MGF |
![{\displaystyle (1-p+pe^{t})^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9bd9001eff48b6efb6779c428c96923648ffc5ca) |
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CF |
![{\displaystyle (1-p+pe^{it})^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4fd07265d9837b2d2e1775565cbec7e24e5c21e5) |
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PGF |
![{\displaystyle G(z)=[(1-p)+pz]^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ad93fbb1208a6637ccd78540a9ddfc2e65a91ee2) |
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Fisher information |
![{\displaystyle g_{n}(p)={\frac {n}{p(1-p)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/808ec49ba969e0331628702ad84f53b3425540d5) (for fixed ) |
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Binomial distribution
Probability mass function ![Probability mass function for the binomial distribution](//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Binomial_distribution_pmf.svg/300px-Binomial_distribution_pmf.svg.png) |
Cumulative distribution function ![Cumulative distribution function for the binomial distribution](//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/300px-Binomial_distribution_cdf.svg.png) | Notation |
![{\displaystyle B(n,p)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/58280f6b0f1a1b474a7047c07943f908e775aa71) |
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Parameters |
– number of trials
– success probability for each trial |
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Support |
– number of successes |
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PMF |
![{\displaystyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e8a14d7d4b8ca69d8164da826f6c0c683254) |
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CDF |
![{\displaystyle I_{1-p}(n-k,1+k)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5e0094516ca80b89715e5ab3c00d033b1958389c) |
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Mean |
![{\displaystyle np}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27) |
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Median |
or ![{\displaystyle \lceil np\rceil }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e7ade49121f3f129664d81676c95c51302a43653) |
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Mode |
or ![{\displaystyle \lceil (n+1)p\rceil -1}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a054d92b493838f6bfe891a53e1629bdf8f85fb4) |
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Variance |
![{\displaystyle np(1-p)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/57f093250a1d822df677a03ac8aa78c6a8029866) |
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Skewness |
![{\displaystyle {\frac {1-2p}{\sqrt {np(1-p)}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a43fb738bf588b4cd915d98195b64b5c274dfe61) |
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Ex. kurtosis |
![{\displaystyle {\frac {1-6p(1-p)}{np(1-p)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a4c21cc7952fe9e6fbe001e2339d9f3d021846d3) |
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Entropy |
![{\displaystyle {\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/51f7c6331e9bb5bc6f03fc5e34f795d4b025ea89) in shannons. For nats, use the natural log in the log. |
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MGF |
![{\displaystyle (1-p+pe^{t})^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9bd9001eff48b6efb6779c428c96923648ffc5ca) |
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CF |
![{\displaystyle (1-p+pe^{it})^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4fd07265d9837b2d2e1775565cbec7e24e5c21e5) |
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PGF |
![{\displaystyle G(z)=[(1-p)+pz]^{n}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ad93fbb1208a6637ccd78540a9ddfc2e65a91ee2) |
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Fisher information |
![{\displaystyle g_{n}(p)={\frac {n}{p(1-p)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/808ec49ba969e0331628702ad84f53b3425540d5) (for fixed ) |
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Side by side comparison{{Infobox probability distribution}} | {{Infobox probability distribution/sandbox}} |
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Geometric
Probability mass function ![](//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Geometric_pmf.svg/450px-Geometric_pmf.svg.png) |
Cumulative distribution function ![](//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Geometric_cdf.svg/450px-Geometric_cdf.svg.png) | Parameters |
success probability (real) |
success probability (real) |
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Support |
k trials where ![{\displaystyle k\in \{1,2,3,\dots \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0b39d5b5ea8766be593310736b018831df38f0a1) |
k failures where ![{\displaystyle k\in \{0,1,2,3,\dots \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fa9dc253300a153c10599f44f4dd13bcb64d9ad2) |
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PMF |
![{\displaystyle (1-p)^{k-1}p}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/bb5e7585453236373eb291ec670d2bf1848b7b7f) |
![{\displaystyle (1-p)^{k}p}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a683821b02a8f9e3b45546ca960cf03c0cde81f4) |
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CDF |
![{\displaystyle 1-(1-p)^{k}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fcae4d88feab20c5f330a45d2fb78803276abfd6) |
![{\displaystyle 1-(1-p)^{k+1}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0f554bb736dbf8ebb3fea37543de3f10bd34abc4) |
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Mean |
![{\displaystyle {\frac {1}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/31025cccb5c9d719d490bfc933e8d7b6b6f2b425) |
![{\displaystyle {\frac {1-p}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4df1ce4dde6cecdcd57fa1ccabd58e9fee92b1a4) |
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Median |
(not unique if is an integer) |
(not unique if is an integer) |
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Mode |
![{\displaystyle 1}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 0}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
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Variance |
![{\displaystyle {\frac {1-p}{p^{2}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1b49d97d4d08bd6f58dbd7a7d4ea63a2e3e53131) |
![{\displaystyle {\frac {1-p}{p^{2}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1b49d97d4d08bd6f58dbd7a7d4ea63a2e3e53131) |
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Skewness |
![{\displaystyle {\frac {2-p}{\sqrt {1-p}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a8a9eda594276861ebdbbfc5b94040f793c52adb) |
![{\displaystyle {\frac {2-p}{\sqrt {1-p}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a8a9eda594276861ebdbbfc5b94040f793c52adb) |
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Excess kurtosis |
![{\displaystyle 6+{\frac {p^{2}}{1-p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9200c5f9448712278fbce96188e0a92119bbfde3) |
![{\displaystyle 6+{\frac {p^{2}}{1-p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9200c5f9448712278fbce96188e0a92119bbfde3) |
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Entropy |
![{\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/021a04c1686b90893daeddcbd604151e83f50a53) |
![{\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/021a04c1686b90893daeddcbd604151e83f50a53) |
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MGF |
![{\displaystyle {\frac {pe^{t}}{1-(1-p)e^{t}}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ea9ed8c14b02bd2701f1d2430e0e036775506b16) for ![{\displaystyle t<-\ln(1-p)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/da5a468d5543be9193476746a86349c0f02a3427) |
![{\displaystyle {\frac {p}{1-(1-p)e^{t}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8514eb8e2d542a51f4f54be9364085a074ac9430) |
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CF |
![{\displaystyle {\frac {pe^{it}}{1-(1-p)e^{it}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/dc6df4e3d0079a04bc6b6e2633112c19d81d0dbe) |
![{\displaystyle {\frac {p}{1-(1-p)e^{it}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/182af26d7740b1429045c1db739fdb08d8412160) |
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Geometric
Probability mass function ![](//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Geometric_pmf.svg/450px-Geometric_pmf.svg.png) |
Cumulative distribution function ![](//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Geometric_cdf.svg/450px-Geometric_cdf.svg.png) | Parameters |
success probability (real) |
success probability (real) |
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Support |
k trials where ![{\displaystyle k\in \{1,2,3,\dots \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0b39d5b5ea8766be593310736b018831df38f0a1) |
k failures where ![{\displaystyle k\in \{0,1,2,3,\dots \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fa9dc253300a153c10599f44f4dd13bcb64d9ad2) |
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PMF |
![{\displaystyle (1-p)^{k-1}p}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/bb5e7585453236373eb291ec670d2bf1848b7b7f) |
![{\displaystyle (1-p)^{k}p}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a683821b02a8f9e3b45546ca960cf03c0cde81f4) |
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CDF |
![{\displaystyle 1-(1-p)^{k}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fcae4d88feab20c5f330a45d2fb78803276abfd6) |
![{\displaystyle 1-(1-p)^{k+1}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0f554bb736dbf8ebb3fea37543de3f10bd34abc4) |
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Mean |
![{\displaystyle {\frac {1}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/31025cccb5c9d719d490bfc933e8d7b6b6f2b425) |
![{\displaystyle {\frac {1-p}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4df1ce4dde6cecdcd57fa1ccabd58e9fee92b1a4) |
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Median |
(not unique if is an integer) |
(not unique if is an integer) |
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Mode |
![{\displaystyle 1}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 0}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
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Variance |
![{\displaystyle {\frac {1-p}{p^{2}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1b49d97d4d08bd6f58dbd7a7d4ea63a2e3e53131) |
![{\displaystyle {\frac {1-p}{p^{2}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1b49d97d4d08bd6f58dbd7a7d4ea63a2e3e53131) |
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Skewness |
![{\displaystyle {\frac {2-p}{\sqrt {1-p}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a8a9eda594276861ebdbbfc5b94040f793c52adb) |
![{\displaystyle {\frac {2-p}{\sqrt {1-p}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a8a9eda594276861ebdbbfc5b94040f793c52adb) |
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Ex. kurtosis |
![{\displaystyle 6+{\frac {p^{2}}{1-p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9200c5f9448712278fbce96188e0a92119bbfde3) |
![{\displaystyle 6+{\frac {p^{2}}{1-p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9200c5f9448712278fbce96188e0a92119bbfde3) |
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Entropy |
![{\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/021a04c1686b90893daeddcbd604151e83f50a53) |
![{\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/021a04c1686b90893daeddcbd604151e83f50a53) |
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MGF |
![{\displaystyle {\frac {pe^{t}}{1-(1-p)e^{t}}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ea9ed8c14b02bd2701f1d2430e0e036775506b16) for ![{\displaystyle t<-\ln(1-p)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/da5a468d5543be9193476746a86349c0f02a3427) |
![{\displaystyle {\frac {p}{1-(1-p)e^{t}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8514eb8e2d542a51f4f54be9364085a074ac9430) |
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CF |
![{\displaystyle {\frac {pe^{it}}{1-(1-p)e^{it}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/dc6df4e3d0079a04bc6b6e2633112c19d81d0dbe) |
![{\displaystyle {\frac {p}{1-(1-p)e^{it}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/182af26d7740b1429045c1db739fdb08d8412160) |
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