Gamma Distribution
\[\begin{aligned} \mathrm{Gam}(\lambda \vert a,b) &= \frac {1}{\Gamma(a)} b^a \lambda^{a-1}\exp(-b\lambda) \end{aligned}\]-
To verity the integral over $\lambda$ is 1
\[\begin{aligned} \int_0^\infty \mathrm{Gam}(\lambda \vert a,b) &= \int_0^\infty \frac {1}{\Gamma(a)} b^a \lambda^{a-1}\exp(-b\lambda)d\lambda \\&= \frac {1}{\Gamma(a)}b^a \int_0^\infty (\frac tb)^{a-1}\exp(-t)d(\frac tb) \\&= \frac {1}{\Gamma(a)} \int_0^\infty t^{a-1}\exp(-t)dt \\ &= \frac {1}{\Gamma(a)} \Gamma(a) \\&= 1 \end{aligned}\] -
The Mean and Variance of Gamma distribution
\[\begin{aligned} \mathbb{E}[\lambda] &= \int_0^\infty \lambda \mathrm{Gam}(\lambda \vert a,b) d\lambda \\&= \int_0^\infty \lambda \frac {1}{\Gamma(a)} b^a \lambda^{a-1}\exp(-b\lambda) d\lambda \\&= \frac {1}{\Gamma(a)} b^a \int_0^\infty (\frac tb)^{a}\exp(-t)d(\frac tb) \\ &= \frac {1}{\Gamma(a)} b^a \frac {1}{b^{a+1}} \Gamma(a+1) \\ &= \frac ab \end{aligned}\] \[\begin{aligned} \mathrm{var}[\lambda] &= \int_0^\infty (\lambda - \frac ab)^2 \mathrm{Gam}(\lambda \vert a,b) d\lambda \\&= \frac {1}{\Gamma(a)} b^a \left( \frac {1}{b^{a+2}} \Gamma(a+2) - \frac {a^2}{b^2} \right) \\ &= \frac {a} {b^2} \end{aligned}\] -
Gamma distribution can be parameterized with just one parameter
-
Let $\lambda = \frac xb$, we can obtain:
\[\begin{aligned} \mathrm{Gam}(\frac xb \vert a,b) &= \frac {1}{\Gamma(a)} b^a (\frac xb)^{a-1}\exp(-b\frac xb) \\&= b \frac {1}{\Gamma(a)} x^{a-1}\exp(-x) \end{aligned}\]-
Define $ \tilde{\mathrm{Gam}}(x \vert a) = \frac {1}{\Gamma(a)} x^{a-1}\exp(-x)$, then we have \(\begin{aligned} \tilde{\mathrm{Gam}}(x \vert a) = \frac 1b \mathrm{Gam}(\frac xb \vert a,b) \end{aligned}\)
- equally, we have: \(\begin{aligned} \mathrm{Gam}(x \vert a,b) = b\tilde{\mathrm{Gam}}(bx \vert a) \end{aligned}\)
-
Then we can see, the one parameter form $\tilde{\mathrm{Gam}}(x \vert a)$ is a scaled verison of the two parameter form $\mathrm{Gam}(\lambda \vert a,b)$ with a scaler $\frac 1b$
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
# Gamma Distributions
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import gamma as gam_fn
from scipy.stats import gamma
def gamma1(x, a, b):
"""Gamma distribution: parameterized with two parameter"""
return 1/gam_fn(a) * b**a * x**(a-1) *np.exp(-b*x)
def gamma2(x,a):
"""Gamma distribution: parameterized with one parameter"""
return 1/gam_fn(a) * x**(a-1) *np.exp(-x)
parameters = [(0.1,0.1), (1,1), (4,6), (10,15)]
fig, axs = plt.subplots(1,4, figsize=(24,6))
for i in range(4):
a, b = parameters[i]
ax = axs[i]
x = np.linspace(gamma.ppf(0.0001, a), gamma.ppf(0.9999, a), 10000)/b
# x = np.linspace(0.01, 1.99, 100)
# ax.plot(x, gamma.pdf(x, a, scale=1/b), 'r', lw=3, alpha=0.8)
# ax.plot(x, gamma1(x,a,b), 'r', lw=3, alpha=0.8)
ax.plot(x, gamma2(b*x,a)*b, 'r', lw=3, alpha=0.8)
ax.set_xlim((0,2))
ax.set_ylim((0,2))
ax.set_title("a={}, b={}".format(a,b))
ax.set_xlabel("$\lambda$")
plt.show()
