import numpy as np
from numpy import sqrt

x = np.matrix([[0.1,0,0,0,0.1],[0,0.4,0,0,0.4],[0,0,0.3,0,0.3],[0,0,0,0.2,0.2],[0.1,0.4,0.3,0.2,1]])
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_15=x[0,4];x_15

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_25=x[1,4];x_25

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
x_35=x[2,4];x_35

x_41=x[3,0];x_41
x_42=x[3,1];x_42
x_43=x[3,2];x_43
x_44=x[3,3];x_44
x_45=x[3,4];x_45

x_51=x[4,0];x_51
x_52=x[4,1];x_52
x_53=x[4,2];x_53
x_54=x[4,3];x_54
x_55=x[4,4];x_55

mu_x=5*x_15+7*x_25+(-4)*x_35+15*x_45
mu_x
mu_y=(-1)*x_51+6*x_52+2*x_53+20*x_54
mu_y
print("Mean of x is : \n", round(mu_x,6))
print("Mean of y is : \n", round(mu_y,6))

var_x=5**2*x_15+7**2*x_25+(-4)**2*x_35+15**2*x_45-mu_x**2
var_x
var_y=(-1)**2*x_51+6**2*x_52+2**2*x_53+20**2*x_54-mu_y**2
var_y
print("Variance of x is : \n", round(var_x,6))
print("Variance of y is : \n", round(var_y,2))

"""
sum_p_xy = ((-5)*x_11+30*x_12+10*x_13+100*x_14
           +(-7)*x_21+42*x_22+14*x_23+140*x_24
           +(4)*x_31+(-24)*x_32+(-8)*x_33+(-80)*x_34     
           +(-15)*x_41+90*x_42+30*x_43+300*x_44)
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))

"""

xx = np.matrix([[0.1,0,0,0],[0,0.4,0,0],[0,0,0.3,0],[0,0,0,0.2]])
p_xy = np.array(xx)

yy= np.matrix([[-5,30,10,100],[-7,42,14,140],[4,-24,-8,-80],[-15,90,30,300]])
xy = np.array(yy)

cov_xy = np.sum(p_xy*xy) - mu_x*mu_y
cov_xy
print("Covariance of x and y is :", round(cov_xy,2))

corr_xy = (cov_xy)/sqrt(var_x*var_y)
corr_xy
print("Correlation of x and y is :", round(corr_xy,3))