- #1
Mogarrr
- 120
- 6
Homework Statement
Write the integral that would define the mgf of the pdf,
[itex]f(x) = \frac 1{\pi} \frac 1{1+x^2} dx [/itex]
Homework Equations
The moment generating function (mgf) is [itex] E e^{tX}[\itex].
The Attempt at a Solution
My question really has to do with improper integrals. I must show the improper integral diverges:
[itex]\int_0^{\infty} e^{tx} \frac 1{\pi} \frac 1{1+x^2} dx [/itex].
Now if do integration by parts and let [itex]u=e^{tx}[/itex] and [itex] dV = \frac 1{1+x^2}dx [/itex], then I have:
[itex] \frac 1{\pi} e^{tx} arctan(x) |_0^{\infty} - \int_0^{\infty} \frac 1{\pi} t arctan(x) e^{tx} dx [/itex].
However I can see that [itex] arctan(x) e^{tx} \frac 1{\pi} |_0^{\infty} [/itex], will be ∞. So is this enough to show that the improper integral diverges?