(1) Recall from last week's homework that the covariance of two random variables is defined by Cov(X,Y) =E[XY] E[X]E[Y]. (a) Show that if E, F C S are two independent events, then Cov(1E, 1f) = 0. (b) Show that if Y, Y2,YN satisfy Cov(Yi, Y;) (ij), then Var(NY;) = N Var(Y) (c) Flip N fair coins and let Xi = {1 th flip H -1 ith flip T = 0 for all pairs (i, j) where for 1iN. Show that for all i j, Cov(Xi, X;) = 0. (d) Use parts (b) and (c) to conclude that if X is the position of a traveler following a random walk after N steps (cf. practice midterm 1) that Var(3x+2) = 9N.