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| cdf = | mean = (see digamma function) | median =No simple closed form | mode = | variance = (see trigamma function) | skewness = | entropy = | mgf = | char = | parameters2 = * α > 0 shape * β > 0 rate | support2 = | pdf2 =〔http://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2006/lecture-notes/lecture6.pdf〕 | cdf2 = | mean2 = (see digamma function) | median2 =No simple closed form | mode2 = | variance2 = (see trigamma function) | skewness2 = | entropy2 = | mgf2 = | char2 = }} In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The common exponential distribution and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use: #With a shape parameter ''k'' and a scale parameter θ. #With a shape parameter ''α'' = ''k'' and an inverse scale parameter β = 1/θ, called a rate parameter. #With a shape parameter ''k'' and a mean parameter μ = ''k''/β. In each of these three forms, both parameters are positive real numbers. The parameterization with ''k'' and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution.〔See Hogg and Craig (1978, Remark 3.3.1) for an explicit motivation〕 The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution〔( ''Scalable Recommendation with Poisson Factorization'' ), Prem Gopalan, Jake M. Hofman, David Blei, arXiv.org 2014〕 – or for that matter, the β of the gamma distribution itself. (The closely related inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.) If ''k'' is an integer, then the distribution represents an Erlang distribution; i.e., the sum of ''k'' independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of 1/θ). The gamma distribution is the maximum entropy probability distribution for a random variable ''X'' for which E() = ''k''θ = α/β is fixed and greater than zero, and E() = ψ(''k'') + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function). ==Characterization using shape ''k'' and scale θ== A random variable ''X'' that is gamma-distributed with shape ''k'' and scale θ is denoted by : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「gamma distribution」の詳細全文を読む スポンサード リンク
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