Binomial distribution
Probability mass function ![Probability mass function for the binomial distribution](//upload.luquay.com/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.luquay.com/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/300px-Binomial_distribution_cdf.svg.png) | Notation |
![{\displaystyle B(n,p)}](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://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e8a14d7d4b8ca69d8164da826f6c0c683254) |
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CDF |
![{\displaystyle I_{1-p}(n-k,1+k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0094516ca80b89715e5ab3c00d033b1958389c) |
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Mean |
![{\displaystyle np}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27) |
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Median |
or ![{\displaystyle \lceil np\rceil }](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://wikimedia.org/api/rest_v1/media/math/render/svg/a054d92b493838f6bfe891a53e1629bdf8f85fb4) |
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Variance |
![{\displaystyle np(1-p)}](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://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/9bd9001eff48b6efb6779c428c96923648ffc5ca) |
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CF |
![{\displaystyle (1-p+pe^{it})^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd07265d9837b2d2e1775565cbec7e24e5c21e5) |
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PGF |
![{\displaystyle G(z)=[(1-p)+pz]^{n}}](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://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.luquay.com/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.luquay.com/wikipedia/commons/thumb/5/55/Binomial_distribution_cdf.svg/300px-Binomial_distribution_cdf.svg.png) | Notation |
![{\displaystyle B(n,p)}](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://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e8a14d7d4b8ca69d8164da826f6c0c683254) |
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CDF |
![{\displaystyle I_{1-p}(n-k,1+k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0094516ca80b89715e5ab3c00d033b1958389c) |
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Mean |
![{\displaystyle np}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27) |
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Median |
or ![{\displaystyle \lceil np\rceil }](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://wikimedia.org/api/rest_v1/media/math/render/svg/a054d92b493838f6bfe891a53e1629bdf8f85fb4) |
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Variance |
![{\displaystyle np(1-p)}](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://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/9bd9001eff48b6efb6779c428c96923648ffc5ca) |
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CF |
![{\displaystyle (1-p+pe^{it})^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd07265d9837b2d2e1775565cbec7e24e5c21e5) |
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PGF |
![{\displaystyle G(z)=[(1-p)+pz]^{n}}](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://wikimedia.org/api/rest_v1/media/math/render/svg/808ec49ba969e0331628702ad84f53b3425540d5) (for fixed ) |
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