Represent the MGF of a rv X by gX (r) = r 0 erxdF(x) + -∞ r ∞ erxdF(x).

In each of the following parts, you are welcome to restrict X to be either discrete or continuous.

(a) Show that the first integral always exists (i.e., is finite) for r ≥ 0 and that the second integral always exists for r ≤ 0.

(b) Show that if the second integral exists for a given r1 > 0, then it also exists for all r in the range 0 ≤ r ≤ r1.

(c) Show that if the first integral exists for a given r2 0, then it also exists for all r in the range r2 ≤ r ≤ 0.

(d) Show that the range of r over which gX (r) exists is an interval from some r- ≤ 0 to some r+ ≥ 0 (the interval might or might not include each endpoint, and the magnitude of either or both endpoints might be 0, ∞, or any point between).

(e) Find an example where r+ = 1 and the MGF does not exist for r = 1. Find another example where r+ = 1 and the MGF does exist for r = 1. Hint: Consider fX (x) = e-x for x  ≥ 0 and figure out how to modify it to fY (y) so that ( ∞ eyfY (y) dy  < >∞ but 0 ey+EyfY (y) =∞ for all E > 0.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.