Problem 1 Suppose that we are using a simplified spherical model of the Earth's surface with latitude u E (5, ") and longitude v E (-7, ); then (if the radius is taken to be 1) the surface area element is given by dA = cosu du du. We will restrict attention to the hemisphere H in which u, v e (-2, 2) -", "). A simple map projection from this hemisphere H is obtained by just taking the x and y coordinates via x = cosu sin v, y = sinu. This is a smooth one-to-one transformation on H. If we pick a point with coordinates (U, V) on H uniformly according to surface area, then the joint density of U and V is fu,v (u, v) = COs 1, 2 (a) Let the longitude v be fixed. Find a formula for the conditional density function fulv=(u). (b) What is the conditional expectation E[|sin U| | V = 0]? (c) Find the joint density of X and Y, where (X, Y) is the image of the random point (U, V) under the map projection defined above. (d) Find a formula for the conditional density function frix=x(v). (e) What is the conditional expectation E[|Y | | X = 0]? (f) Observe that | Y | = |sin U | and the event {X = 0; is exactly the same as the event { V = 0). How is it possible that E [|Y | | X = 0] is not equal to E[|sin U | | V = 0]