I have to prove that the first moment about the origin does not exist. However, I'm having some trouble substantiating the proof. I have some pdf where alpha and beta are two arbitrary constants greater than 0:
$$ f_{Y}(y) = \frac{(\beta / \pi)}{(y-\alpha )^2 + \beta ^2} $$
I need to prove that the first moment does not exist. I know that the expected value has to evaluate to infinity for the moment not to exist. I set up the integral as follows:
$$ E(Y) = \int_{0}^{\infty}\frac{y(\beta / \pi) }{(y-\alpha )^2 + \beta ^2} dy $$
However, I don't know where to go from here. If I could argue that the limit where y approaches infinity reduces the integral down to
$$ E(Y) = \lim_{t\rightarrow \infty} \int_{0}^{t} \frac{1}{y} dy $$
then, by default, the integral diverges and the first moment doesn't exist. Am I allowed to do this or is there an alternative way that I can prove that the first moment about the origin does not exist?
Thanks in advance for all of your help.