this problem is an programming assignment using python.

Problem 1 (Multivariate Normal for Regression) (15-25= 40 points) One interesting application of multivariate normal (MVN) random variables is to solve regression tasks In this problem, you will implement a simple MVN-based predictor that predicts the outputs of the testing queries based on the training data. Specifically, let Direin = {(+1-81). (73-92),... (1x)} be the training dataset and let Dest={N+1+IN+) be the testing queries. The goal is to predict {N+1, 9N+ar) that correspond to {N+IN+1}. (a) As a prep work, please show the following property that we discussed in class: Let 21,2, be a pair of bivariate normal random variable with means, variance 01,03, and correlation coefficient p. Show that conditioned on that Z1 = 51, the conditional distribution of 2 is normal with means tes and variance (b) The property in (a) can be extended to the multivariate normal case. Suppose that for every k E (N + 1.N+). (Y... Yx Y) is multivariate normal with mean vector y = Hall and a (N+ 1) (N + 1) covariance matrix , where the covariance between Y and Y, (denoted by 5 ) has the following form: Diss = a; exp(-(12) + obcy, Vt.) 22 where af is a scalo factor, ( is called the lengthscale, o is some positive constant (usually called the noise parame ter), and is the delta function (i.e. 8, =lifi = j and 6, = 0ifi #1). Given that Yi = 1, Yx = ws.it can be shown that the conditional distribution of Yy is normal with mean K (N)[K()+o1N and variance K(2x) - (*)[K(ANN) +o+/-(.*), where . (**) = 9,1%) is a 1 x N wctor. KELN IN) is an N N matrix with the (1,3)-th entry equal to . ./ is an identity matrix of size NxN. K (PLIN, #a) is the transpose of K ( 121). ..is an N x 1 vector Based on the above conditional distribution, please write a program (e.s. in Python or MATLAB) to find the predictive distributions of the outputs of the test query points [N+Nar). What is the prediction result of the testing dataset under og = 1,0 = 0.1.2=0.5? What is the prediction result of the testing dataset if is set to be 0.1 instead? How about = 2.0