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Gamma distribution 伽马分布
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摘自：https://en.wikipedia.org/wiki/Gamma_distribution 1、描述 In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang 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 （相当于均值）.With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter（相当于均值的倒数）.With a shape parameter k and a mean parameter μ = kθ = α/β. 上述所有参数都是正的实数。2、PDF和CDF的曲线和公式
scale参数即为：指数分布的均值。rate参数即为：指数分布的lambda。较容易混淆的参数，相当重要！！！
The skewness of the gamma distribution only depends on its shape parameter, k, and it is equal to 实际生活相关：for example, 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. If k is a positive 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. 3、伽马分布表示 A random variable X that is gamma-distributed with shape α and rate β is denoted： PDF： 形状α、尺度θ参数 形状α、速率β参数 其中，
CDF： 表中的下不完全伽马函数可以进一步化简（具体化），一般在概率中较常用到：
4、Summation 多个伽马分布的累加和
注意：尺度参数相同，仅形状参数累加 5、Scaling 尺度变换 注意：形状参数不变，仅尺度参数或速率参数被缩放 6、Logarithmic expectation and variance 对数的 In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
也可以写成调和数的关系式：
Hn是第n个调和数，γ是欧拉-马歇罗尼常数。
It is the first of the polygamma functions. 7、Related distributions and properties 相关的其他分布
补充概念： 调和数：the harmonic numbers are defined for positive integers n as
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，α = k
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、
或者
化成1-上不完全伽马函数与伽马函数的关系式，进一步借助于下式：
，即可将CDF表示成累加形式，易于求概率和积分。
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