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Moment Generating Functions

The moment generating function packages an entire distribution into one function:

The moment generating function
MX(t)=E[etX]M_X(t) = E\left[e^{tX}\right]

Differentiate at zero and the moments fall out, which is what makes it a tool rather than a curiosity.

defined for tt in an open interval around 0. Where it exists, it does three jobs that are each hard by other means.

Job one: generating moments

Differentiate and evaluate at zero:

E[Xn]=MX(n)(0)E[X^n] = M_X^{(n)}(0)

The mechanism is the exponential series. Since etX=1+tX+t2X22!+⋯e^{tX} = 1 + tX + \frac{t^2X^2}{2!} + \cdots, taking expectations gives

MX(t)=1+tE[X]+t2E[X2]2!+⋯M_X(t) = 1 + tE[X] + \frac{t^2E[X^2]}{2!} + \cdots

so the moments sit in the Taylor coefficients, and differentiating picks them out one at a time.

For an exponential distribution with rate λ\lambda, MX(t)=λλ−tM_X(t) = \frac{\lambda}{\lambda - t} for t<λt < \lambda. Then M′(0)=1λM'(0) = \frac{1}{\lambda} and M′′(0)=2λ2M''(0) = \frac{2}{\lambda^2}, giving Var(X)=2λ2−1λ2=1λ2\text{Var}(X) = \frac{2}{\lambda^2} - \frac{1}{\lambda^2} = \frac{1}{\lambda^2} with no integration by parts anywhere.

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