Let T be the triangle in the (x, y)-plane bounded by the lines x = 0, y = 0, and x + y = 1; the three vertices of this triangle are the points (0, 0), (1, 0), and (0, 1).

(a) Let U be a random variable with density (pdf) given by fU (u) = 2u , 0 < u < 1 . The line x + y = U intersects the triangle T. Let S be the part of T which lies below this line; S is the triangle with vertices (0, 0), (U, 0), and (0, U). Let W be the area of S. Find the density of the random variable W.

(b) Suppose n random points are distributed uniformly and independently on the triangle T. (The points are also independent of U.) Let J be the number of these points which lie inside the smaller triangle S. Find EJ.

(c) Find Var(J).

(d) Find P(J = i) for i {0, 1, . . . , n}.