Gamma分布与逆Gamma分布

Gamma分布

若随机变量X的密度函数为

⎧ ⎩ ⎨ ⎪ ⎪ λ α Γ ( α ) x α - 1 e - λ x, x \geq 0 0, x < 0 G a m m a Γ (α) = \int + \infty 0 x α - 1 e - x d x

$\begin{array}{l} \left\{ \begin{array}{l} \frac{{{\lambda ^\alpha }}}{{\Gamma (\alpha )}}{x^{\alpha - 1}}{e^{ - \lambda x}},x \ge 0\\ 0,x < 0 \end{array} \right.\\ Gamma\Gamma (\alpha ) = \int_0^{ + \infty } {{x^{\alpha - 1}}{e^{ - x}}dx} \end{array}$

则称X服从Gamma分布，记为X~ $Ga(\alpha ,\lambda )$

不同alpha值对应的Gamma分布密度函数图像

Gamma分布的期望和方差

E (x) = λ α Γ ( α ) \int + \infty 0 x α e - λ x d x = Γ ( α + 1 ) Γ ( α ) 1 λ = α λ E (x 2) = λ α Γ ( α ) \int + \infty 0 x α + 1 e - λ x d x = Γ ( α + 2 ) Γ ( α ) 1 λ 2 = α ( α + 1 ) λ 2 V a r (x) = E (x 2) - [E (x)] 2 = α ( α + 1 ) λ 2 - α 2 λ 2 = α λ 2

$\begin{array}{l} E(x) = \frac{{{\lambda ^\alpha }}}{{\Gamma (\alpha )}}\int_0^{ + \infty } {{x^\alpha }} {e^{ - \lambda x}}dx = \frac{{\Gamma (\alpha + 1)}}{{\Gamma (\alpha )}}\frac{1}{\lambda } = \frac{\alpha }{\lambda }\\ E({x^2}) = \frac{{{\lambda ^\alpha }}}{{\Gamma (\alpha )}}\int_0^{ + \infty } {{x^{\alpha + 1}}} {e^{ - \lambda x}}dx = \frac{{\Gamma (\alpha + 2)}}{{\Gamma (\alpha )}}\frac{1}{{{\lambda ^2}}} = \frac{{\alpha (\alpha + 1)}}{{{\lambda ^2}}}\\ Var(x) = E({x^2}) - {[E(x)]^2} = \frac{{\alpha (\alpha + 1)}}{{{\lambda ^2}}} - \frac{{{\alpha ^2}}}{{{\lambda ^2}}} = \frac{\alpha }{{{\lambda ^2}}} \end{array}$

其中期望式中的第二个等号处分别使用了 $\alpha$ 与 $\alpha+1$ 次分部积分法，与Gamma函数的性质： $\Gamma(\alpha)=(\alpha-1)!$

Gamma分布的特例

G a (1, λ) = E x p (λ) = λ e - λ x G a (n 2, 1 2) = χ 2 (n) = ( 1 2 ) n 2 Γ ( n 2 ) x n 2 - 1 e - 1 2 x

$\begin{array}{l} Ga(1,\lambda ) = Exp(\lambda ){\rm{ = }}\lambda {{\rm{e}}^{ - \lambda x}}\\ Ga(\frac{n}{2},\frac{1}{2}) = {\chi ^2}(n) = \frac{{{{\left( {\frac{1}{2}} \right)}^{\frac{n}{2}}}}}{{\Gamma \left( {\frac{n}{2}} \right)}}{x^{\frac{n}{2} - 1}}{e^{ - \frac{1}{2}x}} \end{array}$

Gamma分布与泊松分布、指数分布的关系
若一段时间[0,1]内事件A发生的次数服从参数为 $\lambda$ 的泊松分布
两次事件发生的时间间隔将服从参数为 $\lambda$ 的指数分布
n次事件发生的时间间隔服从X~ $Ga(\alpha ,\lambda )$ 分布

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逆Gamma分布

若随机变量X的密度函数为：

⎧ ⎩ ⎨ ⎪ ⎪ λ α Γ ( α ) x - α - 1 exp (- λ x), x \geq 0 0, x < 0 G a m m a Γ (α) = \int + \infty 0 x α - 1 e - x d x

$\begin{array}{l} \left\{ \begin{array}{l} \frac{{{\lambda ^\alpha }}}{{\Gamma (\alpha )}}{x^{ - \alpha - 1}}\exp ( - \frac{\lambda }{x}),x \ge 0\\ 0,x < 0 \end{array} \right.\\ Gamma\Gamma (\alpha ) = \int_0^{ + \infty } {{x^{\alpha - 1}}{e^{ - x}}dx} \end{array}$

则称X服从InvGamma分布，记为 X~ $IG(\alpha ,\lambda )$

不同alpha值对应的逆Gamma分布密度函数图像

Gamma分布的期望和方差

E (x) = λ α Γ ( α ) \int + \infty 0 x - α e - λ x d x = Γ ( α - 1 ) Γ ( α ) λ = λ α - 1 E (x 2) = λ α Γ ( α ) \int + \infty 0 x - α + 1 e - λ x d x = Γ ( α - 2 ) Γ ( α ) λ 2 = λ 2 ( α - 1 ) ( α - 2 ) V a r (x) = E (x 2) - [E (x)] 2 = λ 2 ( α - 1 ) ( α - 2 ) - λ 2 ( α - 1 ) 2 = λ 2 ( α - 1 ) 2 ( α - 2 )

$\begin{array}{l} E(x) = \frac{{{\lambda ^\alpha }}}{{\Gamma (\alpha )}}\int_0^{ + \infty } {{x^{{\rm{ - }}\alpha }}} {e^{ - \frac{\lambda }{{\rm{x}}}}}dx = \frac{{\Gamma (\alpha - 1)}}{{\Gamma (\alpha )}}\lambda = \frac{\lambda }{{\alpha - 1}}\\ E({x^2}) = \frac{{{\lambda ^\alpha }}}{{\Gamma (\alpha )}}\int_0^{ + \infty } {{x^{{\rm{ - }}\alpha + 1}}} {e^{ - \frac{\lambda }{{\rm{x}}}}}dx = \frac{{\Gamma (\alpha - 2)}}{{\Gamma (\alpha )}}{\lambda ^2} = \frac{{{\lambda ^2}}}{{(\alpha - 1)(\alpha - 2)}}\\ Var(x) = E({x^2}) - {[E(x)]^2} = \frac{{{\lambda ^2}}}{{(\alpha - 1)(\alpha - 2)}} - \frac{{{\lambda ^2}}}{{{{(\alpha - 1)}^2}}} = \frac{{{\lambda ^2}}}{{{{(\alpha - 1)}^2}(\alpha - 2)}} \end{array}$
其中期望式中的第二个等号处分别使用了

α $\alpha$ 与

α+1 $\alpha+1$ 次分部积分法,与Gamma函数的性质：

Γ(α)=(α−1)! $\Gamma(\alpha)=(\alpha-1)!$

逆Gamma分布特例

I G (α 2, α λ 2) = I n v - χ 2 (α, λ) = ( α λ 2 ) α 2 Γ ( α 2 ) x - α 2 + 1 e - α λ 2 x

$IG(\frac{\alpha }{2},\frac{{\alpha \lambda }}{2}) = Inv - {\chi ^2}(\alpha ,\lambda ){\rm{ = }}\frac{{{{\left( {\frac{{\alpha \lambda }}{2}} \right)}^{\frac{\alpha }{2}}}}}{{\Gamma \left( {\frac{\alpha }{2}} \right)}}{x^{ - \frac{\alpha }{2} + 1}}{e^{ - \frac{{\alpha \lambda }}{{2x}}}}$

Gamma分布与逆Gamma分布
若随机变量X~ $Ga(\alpha ,\lambda )$ ，则 $\frac{1}{X}$ ~ $IG(\alpha ,\lambda )$

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图像Python代码

Gamma分布

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as st
fig=plt.figure(figsize=(18,6))#确定绘图区域尺寸
ax1=fig.add_subplot(1,2,1)#将绘图区域分成左右两块
ax2=fig.add_subplot(1,2,2)
x=np.arange(0.01,15,0.01)#生成数列

z1=st.gamma.pdf(x,0.9,scale=2)#gamma(0.9,2)密度函数对应值
z2=st.gamma.pdf(x,1,scale=2)
z3=st.gamma.pdf(x,2,scale=2)
ax1.plot(x,z1,label="a<1")
ax1.plot(x,z2,label="a=1")
ax1.plot(x,z3,label="a>1")
ax1.legend(loc='best')
ax1.set_xlabel('x')
ax1.set_ylabel('p(x)')
ax1.set_title("Gamma Distribution lamda=2")

y1=st.gamma.pdf(x,1.5,scale=2)#gamma(1.5,2)密度函数对应值
y2=st.gamma.pdf(x,2,scale=2)
y3=st.gamma.pdf(x,2.5,scale=2)
y4=st.gamma.pdf(x,3,scale=2)
ax2.plot(x,y1,label="a=1.5")
ax2.plot(x,y2,label="a=2")
ax2.plot(x,y3,label="a=2.5")
ax2.plot(x,y4,label="a=3")
ax2.set_xlabel('x')
ax2.set_ylabel('p(x)')
ax2.set_title("Gamma Distribution lamda=2")
ax2.legend(loc="best")

plt.show()

逆Gamma分布

from scipy.stats import invgamma
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
a=[4,5,6]
for i in a:
    mean, var, skew, kurt = invgamma.stats(i,scale=2,moments='mvsk')
    x = np.linspace(invgamma.ppf(0.01,i,scale=2),invgamma.ppf(0.99,i,scale=2), 100)#invgamma.ppf
    ax.plot(x, invgamma.pdf(x,i,scale=2,), label="a="+str(i))
    ax.legend(loc="best")
ax.set_xlabel('x')
ax.set_ylabel('p(x)')
ax.set_title("Invgamma Distribution lamda=2")
plt.show()