3.10 Show that if X and Y are independent random variables, X and Y are also uncorrelated. Show that if X and Y follow a bivariate Normal distribution and X and Y are uncorrelated, then X and Y are also independent. Show that if X and Y are dichotomous random variables (that is, X and Y can each assume only one of two possible values such as zero or one) following a joint distribution, and X and Y are uncorrelated, then X and Y must also be independent. This result will be of interest in the latent class model of latent structure analysis (see Chapter 10). Show by example that if X and Y follow a joint distribution, and Y given X follows a Normal distribution, X and Y need not follow a bivariate Normal distribution. This result is of interest, for instance, in the regression model of Chapter 8.