- #1
TelusPig
- 15
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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]