8. Let Anin>o be a sequence of RVs on a common probability space. (a) Give formal definitions of convergence of RVs in probability and of convergence of RVs almost surely. (b) Prove that convergence almost surely implies convergence in probability, i.e. if Xn " Xo as n -+ oo, then also Xn Xo as n -+ 0o. (c) Assume that Xn ~ B(n, X) for n > 1 and some fixed > > 0. Show by direct computation that Xn -> Xo as n -> oo, where Xo ~ P()). (d) Under conditions of part (c), prove the same result (i.c., that Xn d Xo as n -+ 0, Xo ~ P())) using the method of characteristic functions. Hints: (c) That is, demonstrate by a direct computation that, for any fixed k 2 0, one has P(Xn = k) - P(Xo = k) as n - co. (d) The ChF of P()) is given at the end of the paper. [4+ 4+4+4 =16