chapter_4_continuous_probability_distribution.ipynb

Kernel: Python 3

from scipy.stats import uniform
from scipy.stats import expon
from scipy.stats import gamma
from scipy.stats import weibull_min
from scipy.stats import beta
import math

def print_status(exp, var, pmf, cdf):
    print('Exp\t {:.6f}\nVar\t {:.6f}\nPMF\t {:.6f}\nCDF\t {:.6f}'.format(exp, var, pmf, cdf))

Chpater 4 Coutinuous Probability Distribution

4.1 Uniform Distribution

$X\sim U(a, b):f(x;a, b) = \frac{1}{b-a}, a\leq x\leq b$

$E(X) = \frac{a+b}{2}$

$Var(X) = \frac{(b-a)^2}{12}$

# Input
a = 1
b = 2
pdf_variable = [1, 1.75] # [star point, end point]
cdf_variable = 1.5

# Calculate
exp, var = uniform.stats(a, b-a, moments='mv')
pdf = uniform.cdf(pdf_variable[1], a, b-a) - uniform.cdf(pdf_variable[0], a, b-a)
cdf = uniform.cdf(cdf_variable, a, b-a)

# Output
print_status(exp, var, pdf, cdf)

Exp	 1.500000
Var	 0.083333
PMF	 0.750000
CDF	 0.500000

4.2 Exponential Distribution

$X\sim Exp(\lambda):f(x;\lambda) = \lambda e^{-\lambda x}, x>0$

$E(X) = \frac{1}{\lambda}$

$Var(X) = \frac{1}{\lambda^2}$

# Input
lamb = 0.31
pdf_variable = [1, 1.5] # [star point, end point]
cdf_variable = 5

# Calculate
exp, var = expon.stats(scale=1/lamb, moments='mv')
pdf = expon.cdf(pdf_variable[1], scale=1/lamb) - expon.cdf(pdf_variable[0], scale=1/lamb)
cdf = expon.cdf(cdf_variable, scale=1/lamb)

# Output
print_status(exp, var, pdf, cdf)

Exp	 3.225806
Var	 10.405827
PMF	 0.105312
CDF	 0.787752

4.3 Gamma Distribution

$X\sim Gam(k, \lambda):f(x;k, \lambda) = \frac{\lambda^k}{\Gamma(k)}x^{k-1}e^{-\lambda x}, x>0$

Where $\Gamma(k) = \int_{0}^{\infty}x^{k-1}e^{-x}dx, k>0$

$E(X) = \frac{k}{\lambda}$

$Var(X) = \frac{k}{\lambda^2}$

# Input
k = 3
lamb = 1.8
pdf_variable = [1, 1.5] # [star point, end point]
cdf_variable = 3

# Calculate
exp, var = gamma.stats(k, scale=1/lamb, moments='mv')
pdf = gamma.cdf(pdf_variable[1], k, scale=1/lamb) - gamma.cdf(pdf_variable[0], k, scale=1/lamb)
cdf = gamma.cdf(cdf_variable, k, scale=1/lamb)

# Output
print_status(exp, var, pdf, cdf)

Exp	 1.666667
Var	 0.925926
PMF	 0.236997
CDF	 0.905242

4.4 Weibull Distribution

$X\sim Weibull(k, \lambda):f(x;k, \lambda) = k\lambda(\lambda y)^{k-1}e^{-(\lambda y)^k}, k>0, \lambda>0$

$E(X) = \frac{1}{\lambda}\Gamma(1+\frac{1}{k})$

$Var(X) = \frac{1}{\lambda^2}\{\Gamma(1+\frac{2}{k})-[\Gamma(1+\frac{1}{a})]^2\}$

# Input
k = 5
lamb = 2
pdf_variable = [1, 2] # [star point, end point]
cdf_variable = 1

# Calculate
exp, var = weibull_min.stats(k, scale=1/lamb, moments='mv')
pdf = weibull_min.cdf(pdf_variable[1], k, scale=1/lamb) - weibull_min.cdf(pdf_variable[0], k, scale=1/lamb)
cdf = weibull_min.cdf(cdf_variable, k, scale=1/lamb)

# Output
print_status(exp, var, pdf, cdf)

Exp	 0.459084
Var	 0.011057
PMF	 0.000000
CDF	 1.000000

4.5 Beta Distribution

$X\sim Beta(\alpha, \beta):f(x;\alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}, 0<x<1$

$E(X) = \frac{\alpha}{\alpha + \beta}$

$Var(X) = \frac{\alpha\beta}{(\alpha +\beta)^2(\alpha + \beta + 1)}$

# Input
a = 10
b = 4
pdf_variable = [0.2, 0.3] # [star point, end point]
cdf_variable = 0.2

# Calculate
exp, var = beta.stats(a, b, moments='mv')
pdf = beta.cdf(pdf_variable[1], a, b) - beta.cdf(pdf_variable[0], a, b)
cdf = beta.cdf(cdf_variable, a, b)

# Output
print_status(exp, var, pdf, cdf)

Exp	 0.714286
Var	 0.013605
PMF	 0.000636
CDF	 0.000016

Addup: Calculate the Gamma Function

$\Gamma(k) = \int_{0}^{\infty}x^{k-1}e^{-x}dx, k>0$

math.gamma(6)

120.0