
Challenging Problem
Hello math minds,
This is my first post. I was over at physics forums (under a different name) but have quickly migrated to this site after finding it (Rofl).
I would appreciate it if someone could walk me through this, if like me, you find yourself with a lot of free time.
Let $\displaystyle X $ be a random variable with probability density function
$\displaystyle f(x) = (x \sigma \sqrt{2\pi})^{1}) (1 + \varepsilon \ sin (2 \pi n \frac{(log(x)  \mu)}{\sigma})) exp ( \frac{1}{2 \sigma^2} (log(x)  \mu)^2) \ \ \, x>0 $
and $\displaystyle f(x) = 0 \ \ \, x \leq 0 $ (a piecewise function) where $\displaystyle \mu \in R$ $\displaystyle , \sigma > 0$ $\displaystyle , \varepsilon \in [0,1), \ and \ n \in N $ are parametres (N and R denote natural and real numbers). Note that when $\displaystyle \varepsilon = 0, f(x) $ is the pdf corresponding to a longnormal distribution.
(1) Show that $\displaystyle \int_{ \infty}^{\infty} f(u) du = 1 $
(2) Show that the moments $\displaystyle E[X^{\alpha}] $for all $\displaystyle \alpha \in \ R $ do not depend on $\displaystyle \varepsilon $ or n.
(3) Show that $\displaystyle M(t) = E[e^{tX}] $ is infinite for any t > 0
Good Luck

first is obvious
Integral in 1 is just a sum of integral over lognormal density and integral which is reduced (by introducing new variable =( log(x)mu )/ sigma ) to integral over odd function which equals to zero