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## Main Question or Discussion Point

Moment generating functions:

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

Recall:

[itex]M_X(t)=E(e^{tx})=\int_{-\infty}^{\infty}e^{tx}f(x)dx[/itex]

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

[itex]Var(X)=E(X^2)-[E(X)]^2=E[(X-E(X))^2][/itex]

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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?

[itex]\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)[/itex]

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

Recall:

[itex]M_X(t)=E(e^{tx})=\int_{-\infty}^{\infty}e^{tx}f(x)dx[/itex]

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

[itex]Var(X)=E(X^2)-[E(X)]^2=E[(X-E(X))^2][/itex]

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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?

[itex]\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)[/itex]