I want to generate a bivariate normal distribution
\begin{equation}
\begin{pmatrix}
x \\
y
\end{pmatrix}
= N
\begin{pmatrix}
\begin{pmatrix}
\mu_x\\
\mu_y
\end{pmatrix}
_,
\begin{pmatrix}
\sigma_x^2 & \rho\sigma_x\sigma_y \\
\rho\sigma_x\sigma_y & \sigma_y^2 \\
\end{pmatrix}
\end{pmatrix}
\end{equation}
First, generate $ x = N (\ mu_x, \ sigma_x) $. Next, x should be fixed to generate y. The conditional distribution of y at this time is
y|x \sim N(\mu_y + \rho\frac{\sigma_y}{\sigma_x}(x-\mu_x), (1-\rho^2)\sigma_y^2))
Given in.
When implemented in python
python3
import numpy as np
import matplotlib.pyplot as plt
def MVNORM(mu_x, sigma_x, mu_y, sigma_y, rho, N=1):
x = np.random.normal(mu_x, sigma_x, N)
y = np.random.normal(mu_y + rho*sigma_y/sigma_x*(x-mu_x), np.sqrt((1-rho**2)*sigma_y**2), N)
return([x,y])
#Mean 0, standard deviation 9 x and correlation coefficient 0.Generate 1000 y with standard deviation 3 with mean 3 correlated at 9
MV = MVNORM(0, 9, 3, 3, 0.9 ,1000)
plt.scatter(MV[0], MV[1])
plt.show()
SUMMARY = (np.mean(MV[0]), np.std(MV[0]), np.mean(MV[1]), np.std(MV[1]))
print("mean X: {0[0]:0.2f}, stdev X: {0[2]:0.2f}, mean Y: {0[1]:0.2f}, stdev Y: {0[3]:0.2f}".format(SUMMARY))
mean X: 0.14, stdev X: 9.09, mean Y: 2.99, stdev Y: 3.08
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