Question 2. The bivariate normal distribution has the following probability density function f(x,y:1,2,1,2,)=212121exp(2(12)Q) such that Q=(1x1)2+(2y2)22(1x1)(2y2), for any (x,y)R, such that 1R,2R,1>0,2>0 and 11. (a) Prove that XY=yNormal(XY,XY), such that XY=1+21(y2),XY2=(12)12. Accordingly, deduce the distribution of YX=x. (b) Write an R function to implement the two-stage Gibbs sampler for simulating bivariate normal observations. (c) Implement the two-stage Gibbs sampler to simulate N=10,000 random vectors from the bivariate normal with parameters of your choice. Calculate the means, variances and correlation from each sample and compare them to the true values. Plot the scaled histograms along with the corresponding marginal densities. Hint: You may assume without proof that marginally XNormal(1,1) and YNormal(2,2). Question 2. The bivariate normal distribution has the following probability density function f(x,y:1,2,1,2,)=212121exp(2(12)Q) such that Q=(1x1)2+(2y2)22(1x1)(2y2), for any (x,y)R, such that 1R,2R,1>0,2>0 and 11. (a) Prove that XY=yNormal(XY,XY), such that XY=1+21(y2),XY2=(12)12. Accordingly, deduce the distribution of YX=x. (b) Write an R function to implement the two-stage Gibbs sampler for simulating bivariate normal observations. (c) Implement the two-stage Gibbs sampler to simulate N=10,000 random vectors from the bivariate normal with parameters of your choice. Calculate the means, variances and correlation from each sample and compare them to the true values. Plot the scaled histograms along with the corresponding marginal densities. Hint: You may assume without proof that marginally XNormal(1,1) and YNormal(2,2)