1. 2-D Density Tasks. If f (x, y) = 3 [ Are dy, 0 5 x0 0, else Show that: (a) marginal distribution fx (x) is exponential &(1). (b) marginal distribution fy (y) is Gamma Ga(2, 1). (c) conditional distribution f(y|x) is shifted exponential, f(ylx) = de-1(3-2), y > x. (d) conditional distribution f(xly) is uniform u (0, y). 2. Weibull Lifetimes. A lifetime X (in years) of a particular device is modeled by a Weibull distribution f(x|v, 0) = vox"-exp {-0x"} , x 20, with shape parameter v = 3 and unknown rate parameter 0. The lifetimes of X1 = 3, X2 = 4, and X3 = 2 are observed. Assume that an expert familiar with this type of devices suggested an exponential prior on 0 with rate parameter ) = (a) For the prior suggested by the expert, find the posterior distribution of 0. (b) What are the posterior mean and variance? No need to integrate if you recognize to which family of distributions the posterior belongs