Answer appropriately

(c) Suppose X and Y are rv's. Define the rv Z as Z := X + Y. (i) Suppose X ~ Gaussian(3, 4) and Y ~ Gaussian(5, 9). Assume that X and Y are jointly Gaussian with correlation coefficient px,y = 0.5. Then Z ~ Gaussian(uz, oz) is also Gaussian. Determine /z and oz. (ii) Suppose X and Y are independent with X ~ Binomial(10, 0.3) and Y ~ Binomial(20, 0.3). Is Z also binomially distributed? If yes, give its parameters. If no, explain why not.Consider the linear model Y = alX1 + a2X2 +b where the vector ( X1, X2) has a bivariate Gaussian distribution with means #1, #2, variances of, 02, and correlation coefficient p12. Consider the vector (X1, Y). One can show (via some involved calculus) that (X1, Y) has a bivariate Gaussian distribution. Determine its two means, two variances, covariance, and correlation coefficient. Remark: In lecture we considered precisely this problem but under the as- sumption that X] and X2 are independent.5. Let X1, ..., Xn be a random sample from Uniform(0, 0) with an unknown endpoint 0 > 0. We want to estimate the parameter 0. (a) Find the method of moments estimator (MME) of 0. (b) Find the MLE 0 of 0. (c) (R) Set the sample size as 25, do a simulation in R to compare these two esti- mators in terms of their bias and variance. Include a side-by-side boxplot that compares their sampling distributions.Question 27 1 pts Which one of the following statements is correct: Unbiased estimators are always asymptotically unbiased Asymptotically unbiased estimators are always unbiased O Asymptotically unbiased estimators are always consistent Unbiased estimators are always consistent