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Fundamentals of Probability Theory - 15 - Gamma Distribution

2022-08-05 14:32:00 【why why】

This paper documents the gamma distribution.

Gamma distribution of integer degrees

If the event follows a Poisson distribution,The Poisson distribution parameters are \lambda,then the eventi times and timesi+k time interval between occurrencestThe distribution is a gamma distribution.

概率密度函数

p(t ; \lambda, k)=\frac{t^{(k-1)} \lambda^{k} e^{(-\lambda t)}}{\Gamma(k)}

其中 t 为时间间隔.

期望

\mathbb{E}[t]=\frac{k}{\lambda}

方差

\operatorname{Var}[t]=\frac{k}{\lambda^{2}}

上面的定义中 k 必须是整数.

A more general gamma distribution

事实上,若随机变量 X 服从伽马分布,则其概率密度函数为：

p(X ; \alpha, \beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} X^{\alpha-1} e^{-\beta X}, \quad X>0

期望

\mathbb{E}[X]=\frac{\alpha}{\beta}

方差

\operatorname{Var}[X]=\frac{\alpha}{\beta^{2}}

当 \alpha \leq 1 时, p(X ; \alpha, \beta) 为递减函数. 当 \alpha>1 p(X ; \alpha, \beta) 为单峰函数.

Understanding of integer degree gamma distribution

已知Gamma分布的密度函数为:

f(x, \alpha, \lambda)=\frac{\lambda^{\alpha} x^{\alpha-1} e^{-\lambda x}}{\Gamma(\alpha)}, x>0

then its in time (t,+\infty) 上的积分为

即有：

令 \alpha 为 n, \quad n=0,1,2, \cdots , 有

\int_{t}^{+\infty} f(x, n, \lambda) \cdot d x=\int_{t}^{+\infty} f(x, n-1, \lambda) \cdot d x+\frac{(\lambda t)^{n-1} \cdot e^{-\lambda t}}{(n-1) !}

叠加求和, 得:

\int_{t}^{+\infty} f(x, n, \lambda) \cdot d x=\sum_{k=0}^{n-1} \frac{(\lambda t)^{k} \cdot e^{-\lambda t}}{k !}

GammaThe above integral form of distribution can be understood as : 第 n events happen exactly in time (t,+\infty) 的概率, 相当于在时间 (0, t) happens within 0,1,2, \cdots, n-1 The sum of the probabilities of the events.
也可以反过来说,The gamma distribution isnThe sum of independent exponentially distributed random variables.

伽马函数

伽玛函数（Gamma Function）作为阶乘的延拓,是定义在复数范围内的亚纯函数,通常写成 \Gamma(x) . 在xWhen the value is a positive integer, it is unified with the factorial.

在实数域上伽玛函数定义为:

\Gamma(x)=\int_{0}^{+\infty} t^{x-1} e^{-t} \mathrm{~d} t(x>0)

在复数域上伽玛函数定义为:

\Gamma(z)=\int_{0}^{+\infty} t^{z-1} e^{-t} \mathrm{~d} t

其中 \operatorname{Re}(z)>0

除了以上定义之外,There is another way of writing the gamma function formula:

\Gamma(x)=2 \int_{0}^{+\infty} t^{2 x-1} e{-t{2}} \mathrm{~d} t

我们都知道 \int_{0}^{+\infty} e{-t{2}} \mathrm{~d} t=\frac{\sqrt{\pi}}{2} is a common integral result The above formula can be used \Gamma\left(\frac{1}{2}\right)=2 \int_{0}^{\perp \infty} e{-t{2}} \mathrm{~d} t=\sqrt{\pi} 来验证