Now let's move onto the next step, where you will construct the covariance matrix for our data. Recall the following formula for an element of the covariance matrix:

\operatorname{cov}[x,y] = E[(x-\mu_{x})(y-\mu_{y})] \approx \frac{1}{N}\sum_{i=1}^{N}(x_{i}-\mu_{x})(y_{i}-\mu_{y})cov[x,y]=E[(xx)(yy)]N1i=1N(xix)(yiy)

Here, the subscript ii refers to the iith entry of a given variable. \mu_xx represents the mean of xx and \mu_{y}y represents the mean of yy. NN represents the number of data points which for our data set is 16.

The covariance matrix itself is given by:

A=\begin{bmatrix} \operatorname{cov}[x,x] & \operatorname{cov}[x,y] \\ \operatorname{cov}[y,x] & \operatorname{cov}[y,y] \end{bmatrix}.A=[cov[x,x]cov[y,x]cov[x,y]cov[y,y]].

You will calculate the covariance matrix for the data in the previous questions in the code cell below. Remember that data_normalized[i,0] refers to the normalized iith element in xx, given by (x_{i}-\mu_{x})(xix). Similarly, data_normalized[i,1] refers to the normalized iith element in yy, given by (y_{i}-\mu_{y})(yiy). Hence data_normalized[:,0] is a 2D array (column vector) containing all the normalized data for variable xx, and similarly data_normalized[:,1] is a 2D array (column vector) containing all the normalized data for variable yy. The variable cov_xx is equivalent to \operatorname{cov}[x,x]cov[x,x]. Complete the code cell by calculating the remaining elements of the covariance matrix. cov_xy has been provided as an example. When you are done, directly copy & paste the output of the code cell into the answer box below.

# Complete the code below to compute cov[x, x] cov[y, x] and cov[y, y].

# cov_xy has been provided as an example

cov_xx = (1/N)*np.sum(data_normalized[:,0]*data_normalized[:,0])

# This prints the final covariance matrix

# using the elements computed above

# Once you have finished writing the code above,

# DIRECTLY copy the output of the following function

# to the answer cell.

print_covariance_matrix()