1. Suppose X "" N.µ; a 2/. Find the correlation between X and Y , where Y D X 2. Find all values of .µ; a/ for which the correlation is zero.
2. (Maximum Correlation). Suppose .X; Y / has a general bivari- ate normal distribution with a positive correlation p. Show that among all functions g.X/; h.Y / with finite variances, the correlation between g.X/ and h.Y / is maxi- mized when g.X/ D X; h.Y / D Y .
3. (Bivariate Normal Calculation). Suppose X "" N.0; 1/ and, given X D x, Y "" N.x C 1; 1/.
(a) What is the marginal distribution of Y ?
(b) What is the correlation between X and Y ?
(c) What is the conditional distribution of X given Y D y?