import numpy as np
from numpy import sqrt

x = np.matrix([[1/6, 1/12, 1/12, 1/3], [1/12, 1/2, 1/12, 2/3], [1/4, 7/12, 1/6, 1]])
x
print("Matrix is : \n", x)
print(type(x))

x_11=x[0,0];x_11
x_12=x[0,1];x_12
x_13=x[0,2];x_13
x_14=x[0,3];x_14
x_21=x[1,0];x_21
x_22=x[1,1];x_22
x_23=x[1,2];x_23
x_24=x[1,3];x_24
x_31=x[2,0];x_31
x_32=x[2,1];x_32
x_33=x[2,2];x_33
x_34=x[2,3];x_34

mu_x=1*x_14+3*x_24
mu_x
mu_y=0*x_31+1*x_32+2*x_33
mu_y
print("Mean of x is : \n", round(mu_x,6))
print("Mean of y is : \n", round(mu_y,6))

var_x=1**2*x_14+3**2*x_24-mu_x**2
var_x
var_y=0**2*x_31+1**2*x_32+2**2*x_33-mu_y**2
var_y
print("Variance of x is : \n", round(var_x,6))
print("Variance of y is : \n", round(var_y,6))

sum_p_xy = 0*x_11+1*x_12+2*x_13+0*x_21+3*x_22+6*x_23 
sum_p_xy
print("Sum of multiplication of p_xy and xy is : \n", sum_p_xy)

cov_xy = sum_p_xy - mu_x*mu_y
cov_xy
print("Covariance of x and y is :", round(cov_xy,6))

corr_xy = (cov_xy)/sqrt(var_x*var_y)
corr_xy
print("Correlation of x and y is :", round(corr_xy,6))