Question: Momentsofadistributioncanbederivedbydifferentiatingthe moment generatingfunction (mgf ), m(t) = EetY . This functionprovidesanalternativewaytospecifyadistribution. (a) Showthatthe kth derivative m(k)(t) = EY ketY , andhence m(0) = E(Y )

Momentsofadistributioncanbederivedbydifferentiatingthe moment generatingfunction

(mgf ), m(t) = E‰etY Ž.

This functionprovidesanalternativewaytospecifyadistribution.

(a) Showthatthe kth derivative m(k)(t) = E‰Y ketY Ž, andhence m′(0) = E(Y ) and m′′(0) =

E(Y 2).

(b) Showthatthe mgf is m(t) = 1 + tE(Y ) + t2 2!E(Y 2) + t3 3!E(Y 3) + ⋯.

(c) Show thatthe mgf for thePoissondistributionis m(t) = exp[μ(et − 1). Useittofindthe mean andvariance.

(d) The mgf for thenormaldistributionis m(t) = eμt+σ2t2~2. Useittofindthemeanand variance.

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