maxdevblock

beta_hat_1 = sigma_xy / sigma2_x
beta_hat_0 = y_bar - beta_hat_1*x_bar

Y_hat = beta_hat_0 + beta_hat_1*X

sigma2_y_hat = ((Y_hat - y_bar)**2).sum()/Y.size

R2 = sigma2_y_hat / sigma2_y

r2 = r**2

maxdevblock

sigma2_x = ((X - x_bar)**2).sum()/X.size
sigma2_y = ((Y - y_bar)**2).sum()/Y.size

sigma_x = np.sqrt(sigma2_x)
sigma_y = np.sqrt(sigma2_y)

sigma_xy = (X*Y).sum()/X.size - x_bar*y_bar

r = sigma_xy / (sigma_x*sigma_y)

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x_bar = X.sum()/X.size
y_bar = Y.sum()/Y.size

fig, ax = plt.subplots(facecolor="w")
plt.plot(X, Y, "o", label="observations")
plt.axvline(x_bar, ls="--", label=fr"$\bar{{x}}={x_bar:.2f}$", color="C1")
plt.axhline(y_bar, ls="--", label=fr"$\bar{{y}}={y_bar:.2f}$", color="C2")
plt.legend(loc="upper left", bbox_to_anchor=(1,1))
plt.show()

maxdevblock

import numpy as np
import matplotlib.pyplot as plt

X = np.arange(20).astype(float)
Y = X*2
X += np.random.random(size=X.size)*5
Y += np.random.random()*5

plt.plot(X, Y, "o", label="observations")
plt.legend(loc="upper left", bbox_to_anchor=(1,1))

maxdevblock

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

tau = 5
beta = 1/tau
alpha = 5
t = 15
lam = beta*t

maxdevblock

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

tau = 5
beta = 1/tau
alpha = 5
t = 15
lam = beta*t

maxdevblock

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

tau = 5
beta = 1/tau
alpha = 5
d = sps.gamma(a=alpha, scale=1/beta)
x = np.linspace(0, d.ppf(.999), 1000)
y = d.pdf(x)

maxdevblock

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

tau = 5
beta = 1/tau
d = sps.expon(scale=1/beta)
x = np.linspace(0, d.ppf(.999), 1000)
y = d.pdf(x)
P = d.cdf(15)

maxdevblock

# Zn variable from Yn
Zn2 = (Yn - d.mean()) / (d.std()/np.sqrt(n))

# Standard Normal distribution of Zn
plt.hist(Zn2, density=True)
Znd = sps.norm(loc=0, scale=1)
Znx = np.linspace(Znd.ppf(.0001), Znd.ppf(.9999), 100)
Zny = Znd.pdf(Znx)
plt.plot(Znx, Zny, lw=5)
plt.title("Distribution of r.v. $Z_n$ from $Y_n$")

maxdevblock

# Zn variable from Sn
Zn1 = (Sn - n*d.mean()) / (np.sqrt(n)*d.std())

# Standard Normal distribution of Zn
plt.hist(Zn1, density=True)
Znd = sps.norm(loc=0, scale=1)
Znx = np.linspace(Znd.ppf(.0001), Znd.ppf(.9999), 100)
Zny = Znd.pdf(Znx)
plt.plot(Znx, Zny, lw=5)
plt.title("Distribution of r.v. $Z_n$ from $S_n$")