10. Multivariate normal distribution

• This task illustrates an important relationship between the multidimensional and the one-dimensional normal distribution. The task is to produce 1000 points of 2-dimensional normal distribution with a given expected value and given covariance matrix and plot them on the plane. The figure has to be done in two ways! First, with the linear transformation of a random vector variable formed by two independent normal distributed random variable. For this you may use the function normal of the module numpy.random (and e.g. numpy.linalg). If \(\mathbf M\) denotes the transformation matrix, then the covariance matrix is \(\mathbf{M}\mathbf{M}^T\). Second, we generate random points by using the multivariate_normal function of the module numpy.random. One of the arguments of this function is the covariance matrix. Finally, draw the ellipse that contains most of the dots of the figure. The semi-axes of the ellipse be 6 times the singular values ​​of the transformation matrix (the singular value here coincides with the square root of the eigenvalue of the covariance matrix). For simplicity, the expected value should be 0 and \((0,0)\).
• Write a program called normal.py which can be run the program from command line. The arguments are the 4 elements of the transformation matrix. For example, the input line

  python3 normal.py 2 5 3 0
  

  means that the transformation matrix is \[ \begin{bmatrix} 2 & 5\\ 3 & 0 \end{bmatrix}, \] and the result of the program will be two figures like the following:
  
• Drawing the ellipses you need the lenght of the axes and the angle of the big axis with the horizontal \(x\)-axis. For this you may use the rad2deg and the arctan functions of the np module. A little help (not necessary to follow):

      from matplotlib.patches import Ellipse    # modul for drawing
  
      ax = plt.subplot(aspect='equal')          # subfigure
      ell = Ellipse(xy=(0,0),                   # fill in!
                    width=...,
                    height=...,
                    edgecolor='r',
                    angle=... )    
      ell.set_facecolor('none')                 # be empty
      ax.add_artist(ell)
      plt.plot(x, y, 'x')                       # these are the dots
      plt.axis('equal')
      plt.show()
  

• The aim of the exercise is to demonstrate the relationship between the covariance matrix of a 2-dimensional normal distribution and the linear transformation of a vector variable of two independent normal distributions.