Recall that a random variable in is bivariate normal or Gaussian if is normal for all . Let be bivariate normal.
(a) (i) Show that if is a real matrix then is bivariate normal.
(ii) Let and . Find the moment generating function of and deduce that the distribution of a bivariate normal random variable is uniquely determined by and .
(iii) Let and for . Let be the correlation of and . Write down in terms of some or all of and . If , why must and be independent?
For each , find . Hence show that for some normal random variable in that is independent of and some that should be specified.
(b) A certain species of East Anglian goblin has left arm of mean length with standard deviation , and right arm of mean length with standard deviation . The correlation of left- and right-arm-length of a goblin is . You may assume that the distribution of left- and right-arm-lengths can be modelled by a bivariate normal distribution. What is the probability that a randomly selected goblin has longer right arm than left arm?
[You may give your answer in terms of the distribution function of a random variable . That is, .J