Gamma分布¶

定义¶

若随机变量 $X$ 的密度函数为 $$ \begin{aligned} &\left\{\begin{array}{l} \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x}, x \geq 0 \\ 0, x<0 \end{array}\right. \\ &\operatorname{Gamma\Gamma }(\alpha)=\int_0^{+\infty} x^{\alpha-1} e^{-x} d x \end{aligned} $$ 则称X服从 Gamma分布，记为X $G a(\alpha, \lambda)$

期望和方差¶

$$ \begin{aligned} &E(x)=\frac{\lambda^\alpha}{\Gamma(\alpha)} \int_0^{+\infty} x^\alpha e^{-\lambda x} d x=\frac{\Gamma(\alpha+1)}{\Gamma(\alpha)} \frac{1}{\lambda}=\frac{\alpha}{\lambda} \\ &E\left(x^2\right)=\frac{\lambda^\alpha}{\Gamma(\alpha)} \int_0^{+\infty} x^{\alpha+1} e^{-\lambda x} d x=\frac{\Gamma(\alpha+2)}{\Gamma(\alpha)} \frac{1}{\lambda^2}=\frac{\alpha(\alpha+1)}{\lambda^2} \\ &\operatorname{Var}(x)=E\left(x^2\right)-[E(x)]^2=\frac{\alpha(\alpha+1)}{\lambda^2}-\frac{\alpha^2}{\lambda^2}=\frac{\alpha}{\lambda^2} \end{aligned} $$

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：「 $(\alpha)=(\alpha-1)$ !

特例¶

$$ \begin{aligned} &G a(1, \lambda)=\operatorname{Exp}(\lambda)=\lambda \mathrm{e}^{-\lambda x} \\ &G a\left(\frac{n}{2}, \frac{1}{2}\right)=\chi^2(n)=\frac{\left(\begin{array}{l} 1 \\ z \end{array}\right)^n}{\Gamma\left(\frac{n}{2}\right)} x z^n-1 e^{-\frac{1}{z^x}} \end{aligned} $$

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
  • $\mathrm{n}$ 次事件发生的时间间隔服从 $x \sim G a(\alpha, \lambda)$ 分布

逆Gamma分布¶

定义¶

若随机变量 $x$ 的概率密度函数为: $\left\{\begin{array}{l}\frac{\lambda^\alpha}{\Gamma(\alpha)} x^{-\alpha-1} \exp \left(-\frac{\lambda}{x}\right), x \geq 0 \\ 0, x<0\end{array}\right.$ $\operatorname{Gamma} \Gamma(\alpha)=\int_0^{+\infty} x^{\alpha-1} e^{-x} d x$ 则称X服从逆Gamma分布，记作: $x \sim I G(\alpha, \lambda)$

期望及方差¶

$$ \begin{aligned} &E(x)=\frac{\lambda^\alpha}{\Gamma(\alpha)} \int_0^{+\infty} x^{-\alpha} e^{-\frac{\lambda}{\mathrm{x}}} d x=\frac{\Gamma(\alpha-1)}{\Gamma(\alpha)} \lambda=\frac{\lambda}{\alpha-1} \\ &E\left(x^2\right)=\frac{\lambda^\alpha}{\Gamma(\alpha)} \int_0^{+\infty} x^{-\alpha+1} e^{-\frac{\lambda}{\mathrm{x}}} d x=\frac{\Gamma(\alpha-2)}{\Gamma(\alpha)} \lambda^2=\frac{\lambda^2}{(\alpha-1)(\alpha-2)} \\ &\operatorname{Var}(x)=E\left(x^2\right)-[E(x)]^2=\frac{\lambda^2}{(\alpha-1)(\alpha-2)}-\frac{\lambda^2}{(\alpha-1)^2}=\frac{\lambda^2}{(\alpha-1)^2(\alpha-2)} \end{aligned} $$

特例¶

若随机变量 $x \sim G a(\alpha, \lambda)$, 则 $\frac{1}{x} \sim I G(\alpha, \lambda)$

参考资料

  • https://blog.csdn.net/qq_42324085/article/details/118682965
  • https://www.its203.com/article/weixin_41875052/79843374
In [ ]:
In [2]:
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()
In [ ]: