In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family,[1] has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.[2]
Quick Facts Parameters, Support ...
Conway–Maxwell–Poisson
Probability mass function |
Cumulative distribution function |
Parameters |
![{\displaystyle \lambda >0,\nu \geq 0}](//wikimedia.org/api/rest_v1/media/math/render/svg/a4e8272a523205a32eb7faffdff6dcbdeae6bc06) |
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Support |
![{\displaystyle x\in \{0,1,2,\dots \}}](//wikimedia.org/api/rest_v1/media/math/render/svg/687c7daa942b54b74580768fb1f139ddf3b01da2) |
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PMF |
![{\displaystyle {\frac {\lambda ^{x}}{(x!)^{\nu }}}{\frac {1}{Z(\lambda ,\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/414a7bd0a770dfbd91717f805b240393e88990a3) |
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CDF |
![{\displaystyle \sum _{i=0}^{x}\Pr(X=i)}](//wikimedia.org/api/rest_v1/media/math/render/svg/4156ea05351413673e30ebcfccb25fcfae6bbca7) |
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Mean |
![{\displaystyle \sum _{j=0}^{\infty }{\frac {j\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ccbfcd749131801d7bf223866e8d82dbb97a901c) |
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Median |
No closed form |
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Mode |
See text |
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Variance |
![{\displaystyle \sum _{j=0}^{\infty }{\frac {j^{2}\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}-\operatorname {mean} ^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/eba87841753adfa89f2c3f620f502ab1a62c9fad) |
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Skewness |
Not listed |
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Excess kurtosis |
Not listed |
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Entropy |
Not listed |
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MGF |
![{\displaystyle {\frac {Z(e^{t}\lambda ,\nu )}{Z(\lambda ,\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ca803811d02ca59e59e733115125d76cc7feaeb2) |
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CF |
![{\displaystyle {\frac {Z(e^{it}\lambda ,\nu )}{Z(\lambda ,\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/03dd9e79d08a65e0d62d9b8d877d90581ddb564c) |
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PGF |
![{\displaystyle {\frac {Z(t\lambda ,\nu )}{Z(\lambda ,\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/506cb9fa35910ef6700f18efa6d5d224d5b95d53) |
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