Question

value when p= p= p= 1, and p = h. Note that the kurtosis increases as p decreases. 1.9.17. Let y(t) = log M(t), where M(t) is the mgf of a distribution. Prove that V'(0) = pi and "(0) = 02. The function f(t) is called the cumulant generating function.

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Answer 1

ANSWER:

he momen cumulant Generating function A cumulant genesaling function fakes the moment of a poobeba -lity denily function and generates the cumulant- Let Yl)= log ML) Prove that y"Co)-44 4"(0) -72. Prof' for Nandom Variable x with probability denisły funcion fta), by definition of moment generating function Me lit) = Eletx] we have that YIA) - logMce) = log Eletx Derivative «lt) thus 4't)= I log Ele] - (to € (etx)] Eletx) - MCH) MCH)

And for to- 4'(o)= M'lo) MCo) - E(x) Ele) -4 the second derivale of the function yle) is Wit) est |M'(E) ML)- (N°4) ? (MCH))? for to using the formula of mihi moment of distribution, MM/C) = E(ym) Eve can obtain that 4'0) 'C) MO) – (M012 (M601)2 = f(x3) Eleo)-(ECX)}? (Efeo))2 = 6(x2)x1- (EX) = {(x2)-42 =82 show that y'(o) = 1 and 4"(0)=82.