1. Multivariate Distributions (25 points) Please answer the following questions about the distribution p(x, y) plotted below. (a) Estimate the mode(s). (b) Estimate the mean. (c) Sketch the marginal distribution for x, p(x). (d) Sketch the conditional distribution for x given y = : 2, p(x | y = 2). (e) Are the variables x and y correlated? Explain why or why not. 6 5 4 > 3 2 1 0 0 1 2 3 4 X 50 6 0.208 0.169 0.130 0.091 0.052 0.013 Figure 1: Distribution obtained by averaging two Gaussian peaks at (2,2) and (4,4) with standard deviations of 0.6 and 1 respectively.