TEAM PROJECT. Moment Generating Function. The moment

Chapter 24, Problem 24.7

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TEAM PROJECT. Moment Generating Function. The moment generating function G(t) is defined by tX ~ tx G(t) = E(e J) = .L.J e 'f(xj) or G(t) = E(etx) = fX ett:f(x) dx where X is a discrete or continuous random variable,respectively.(a) Assuming that termwise differentiation anddifferentiation under the integral sign are (b) Shov. that the binomial distribution has themoment generating function permissible,show that E(Xk) = dkl(O), where d k ) = dkG/dtk. inparticular, /L = G' (0).(c) Using (b), prove (3). (d) Prove (4)(e) Show that the Poisson distribution has the momentgenerating function G{t) = e-lLelLe' and prove (6). (f) Prove x (~) = M (~ = !) . Using this. prove (9).