Distribution
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Moment-generating function MX(t)
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Characteristic function φ(t)
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PDF/PMF
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CDF
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Mean
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Variance
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Bernoulli
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Geometric
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, for
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Binomial B(n, p)
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np
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np(1 − p)
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Poisson Pois(λ)
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--or--
(for where is the Incomplete gamma function and is the floor function)
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Uniform (continuous) U(a, b)
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Uniform (discrete) U(a, b)
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Normal N(μ, σ2)
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μ
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Chi-squared χ2k
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k
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2k
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Gamma Γ(k, θ)
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![{\displaystyle \scriptstyle \operatorname {E} [X]=k\theta \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bcd0564c15fa09e51ad6cef42d662ef6d1dca35)
![{\displaystyle \scriptstyle \operatorname {E} [\ln X]=\psi (k)+\ln(\theta )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade84f1fd2e20c78ef0cf51012ad6626335092c2) (see digamma function)
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![{\displaystyle \scriptstyle \operatorname {Var} [X]=k\theta ^{2}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333617e4e5bd65bdbe335c76c1a7d0327573320e)
![{\displaystyle \scriptstyle \operatorname {Var} [\ln X]=\psi _{1}(k)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7633714acaea7355cb1750b03b1c68d8053d6196) (see trigamma function )
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Exponential Exp(λ)
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λ e−λx
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1 − e−λx
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λ−1
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λ−2
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Multivariate normal N(μ, Σ)
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![{\displaystyle (2\pi )^{-{\frac {k}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }})'{\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3dc526d480769c28a0e99ec430a55e5bee7faf2) exists only when Σ is positive-definite
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(no analytic expression)
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μ
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Σ
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Degenerate δa
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Laplace L(μ, b)
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μ
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2b2
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Negative Binomial NB(r, p)
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involving a binomial coefficient
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the regularized incomplete beta function
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Cauchy Cauchy(μ, θ)
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does not exist
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undefined
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undefined
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