# 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

The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...

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) Show that the Poisson distribution has the moment generating function G(t) = e^{-Î¼}e^{Î¼et} and prove (6).

(d)

Using this, prove (9).

DistributionThe word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...

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