Australian crab data: The les h1uecrab.dat and orangecrab . dat con tain measurements of body depth (Y1) and rear width (Y3), in millimeters, made on 50 male crabs from each of two species, blue and orange. 'We will model these data using a bivariate normal distribution. a) h) For each of the two species, obtain posterior distributions of the pop ulation mean I9 and covariance matrix E as follows: Using the semi conjugate prior distributions for 6' and E, set #0 equal to the sample mean of the data= AD and SD equal to the sample covariance matrix and v.) = :1. Obtain 10,000 posterior samples of I9 and Z. Note that this \"prior\" distribution loosely centers the parameters around empir ical estimates based on the observed data [and is very similar to the unit information prior described in the previous exercise). It cannot be considered as our true prior distribution, as it was derived from the observed data. However, it can be roughly considered as the prior distribution of someone with weak but unbiased information. Plot values of 6' = (61, 9231' for each group and compare. Describe any size differences between the two groups. From each covariance matrix obtained from the Gibbs sampler, ob- tain the corresponding correlation coefcient. From these values, plot posterior densities of the correlations pblue and parange for the two groups. Evaluate differences between the two species by comparing these posterior distributions. In particular, obtain an approximation to Pr(pb.,,e