1. (Generating Function and Moments). Suppose X has the generat- ing function G.s/. Express the variance and the third moment of X in terms of G.s/ and its derivatives.
2. Suppose G.s/ and H.s/ are both generating functions. Show that pG.s/ C .1 - p/H.s/ is also a valid generating function for any p in .0; 1/. What is an interesting interpretation of the distribution that haspG.s/ C .1 - p/H.s/ as its generating function?
3. (Convexity of the MGF). Suppose X has the mgf .t/, finite in some open interval. Show that .t/is convex in that open interval.
4. Find the first four moments, the first four central moments, and the first four cumulants of X, where X is the number of heads obtained in three tosses of a fair coin, and verify all the interrelationships between them stated in the text.
5. * (Cumulants of a Bernoulli Variable). Suppose X has the pmf P.X D 1/ D p; P.X D 0/ D 1 - p. What are the first four cumulants of X?
6. Suppose X has a symmetric distribution P.XD1D p; P.X D 0/ D 1 - 2p. What are its first four cumulants?