## Moment Generating Function (MGF)

a. For the Binomial Distribution,

Mx(t) = E(eXt) = XX) eX‘0X (1 – 0)

X=0

-X

= [(1 – 0) + 0e‘]

where the last equality uses the binomial expansion (a + b)n =

P ( v I aXbn-X with a = 0e‘ and b = (1 – 0). This is the fundamental x=o X /

relationship underlying the binomial probability function and what makes it a proper probability function. b. For the Normal Distribution,

completing the square

Mx(t) = 1 f °° e-222{[x-(^+ta2)]2-(^+ta2)2+^1 dx

os/2rr J-i

_ e"2O2 [^2-^2-2^ta2-tV]

The remaining integral integrates to 1 using the fact that the Normal density is proper and integrates to one. Hence Mx(t) = e^*+ 2°2*2 after some cancellations.

c. For the Poisson Distribution,

= e-X X ^ _ X = eA(e‘-!)

^ X!

X=0

X

where the fifth equality follows from the fact that X = ea and...

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