(3) Let X be a random variable, Z := e|X| and assume that C := =...

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(3) Let X be a random variable, Z := e|X| and assume that C := = E[Z] < . For a (-1, 1) we define the number g(a) := E(ex). Recall that for any real numbers a, u with |a| < 1 one has ueu dy| alelu. leau - 1| = (a) (2 Points) Show that for any a (1,1) and that when a 0 and that lim a0 ca g(a) C (b) (2 Points) Show that for a (-1, 1) with a 0 one has eX 1 (c) (2 Points) Show that g(0) = 1. ex a g(a) - g(0) a 1 X (3) Let X be a random variable, Z := e|X| and assume that C := = E[Z] < . For a (-1, 1) we define the number g(a) := E(ex). Recall that for any real numbers a, u with |a| < 1 one has ueu dy| alelu. leau - 1| = (a) (2 Points) Show that for any a (1,1) and that when a 0 and that lim a0 ca g(a) C (b) (2 Points) Show that for a (-1, 1) with a 0 one has eX 1 (c) (2 Points) Show that g(0) = 1. ex a g(a) - g(0) a 1 X