Problem 3. An interesting property of Student's & distribution is that the moment generating function does not exist since only finitely many moments exist (for the moment generating function to exist we would need infinitely many moments to exist). a) Once again, the density of the Student's & distribution with d degrees of freedom is given by -(d+1)/2 fr(t) = ² (₁+²) LER. r((d+1)/2) T(d/2)√dn Compute the expectation of T. Be careful to state for what restriction on powers (which will depend on the parameter d) this integral converges. b) Verify that your restriction from part a) implies that the expectation only exists for d > 1. e) Consider the nth moment of T for n > 0. Do not calculate this moment directly (which can be done, but is tedious!), but show for what powers the integral will converge. Note that only the large-t behavior will affect the convergence, so this can be done immediately! By part c), we can conclude that only the first finitely many moments exist, therefore the moment generating function does not exist for the distribution!