Question: Let X be a scalar random variable with moments r and cumulants r in the notation of Section 2.5. Show that the rth moment of

Let X be a scalar random variable with moments μr and cumulants κr in the notation of Section 2.5. Show that the rth moment of the polynomial Y = P (X) = a0 + a1X + a2X2 + …

is given formally by P(d)MX(ξ)∣ξ=0 where d = d/dξ. Hence show that the moment generating function of Y is MY (ς) = exp {ςP (d)}MX (ξ)|ξ=0

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