# The moment generating function G(t) is defined by or where X is a discrete or continuous random variable, respectively. (a) Assuming that term wise differentiation and differentiation under the integral sign are permissible, show that E(X k ) = G (k) (0), where G (k) = d k G/dt k , in particular, Î¼ = G'(0). (b) Prove (4). (c)

Chapter 24, PROBLEM SET 24.7 #16
The moment generating function G(t) is defined by or where X is a discrete or continuous random variable, respectively.

(a) Assuming that term wise differentiation and differentiation under the integral sign are permissible, show that E(Xk) = G(k)(0), where G(k) = dkG/dtk, in particular, Î¼ = G'(0).

(b) Prove (4).

(c) Show that the Poisson distribution has the moment generating function G(t) = e-Î¼eÎ¼et and prove (6).

(d) Using this, prove (9).

Distribution
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