Question

Moment generating functions:
How can I show that \[Var(X)=\frac{d^2}{dt^2}ln M_X(t)\big |_{t=0}\]

Answer 1

Recall:
\(M_X(t)=E(e^{tx})=\int_{-\infty}^{\infty}e^{tx}f(x)dx\)

\(E(X^n)=\frac{d^n}{dt^n}M_X(t)\big |_{t=0}\)

\(Var(X)=E(X^2)-[E(X)]^2=E[(X-E(X))^2]\)
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I tried just applying the equation given but I don't know what to do with the log of this general integral?
\[\frac{d^2}{dt^2}ln M_X(t)=\frac{d^2}{dt^2}ln \left( \int_{-\infty}^{\infty}e^{tx}f(x)dx) \right)\]

Answer 2

Ah nvm the proof is way easier than I thought, and no integrals involved :)