STAB52H3 Introduction to Probability

7. Let X1, X2, ... be i.i.d. random variables such that P(X; = 1) = P(Xi = -1) = ?. (a) (4 points) Show carefully that the moment generating function of X; is given by M (t) = ? (et test ), tER. (b) (8 points) You are given that for t ( R we have the inequality ?(et te t) 0 and t > 0 we have the inequality P( Zn > >) 0 be fixed. By (b), since P(Zn > x) 0, we have P(Zn 2 2) 0. By finding the minimum value of the right hand side, show that for > > 0 we have the upper bound P(Zn > z) 1