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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 ?

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