1 Terms

\[P(X=\text{"it will rain today"})\]
\(X\) is the random variable
\(\mu\) mu is the mean
\(\sigma\) sigma is std (z-score = sigma) \(\sigma^2\) sigma squared is variance

2 Probability density function (pdf)

\[pdf(x;\mu , \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}}exp({-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}})\]

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats

mu = 0
sigma = 1
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
y = stats.norm.pdf(x, mu, sigma) #stats lib pdf

plt.plot(x,y )
plt.show()

mu = 0 #mean
sigma = 1 #sigma-squared is variance
pdf = lambda x: (1/(np.sqrt(2*np.pi*sigma**2)))*(np.e**((-0.5*(mu-x)**2)/(sigma**2)))
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
y = list(map(pdf,x))
                                        
plt.plot(x,y)
plt.show()

3 CDF: Integrate PDF

• CDF :: z-score -> percentile
  • in logistic regression we pretend percentile is probability

\[cdf(x;\mu , \sigma^2) = \int_{-\infty}^{x} pdf(x;\mu , \sigma^2) \partial x \]

import scipy.integrate as integrate
cdf = lambda x: integrate.quad(f,-np.inf,x)[0]
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
y =list(map(cdf,x))
plt.plot(x,y)

std1 = integrate.quad(f,-1,1)
std2 = integrate.quad(f,-2,2)
std3 = integrate.quad(f,-3,3)
full = integrate.quad(f,-np.inf,np.inf)
print(std1,std2,std3,full)

(INTEGRAL         , ERROR-BOUND          )
(0.682689492137086, 7.579375928402476e-15) 
(0.9544997361036417, 1.8403560456416157e-11) 
(0.9973002039367399, 1.1072256503105314e-14) 
(0.9999999999999997, 1.017819138786878e-08)