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Thread: Challenging Problem

  1. #1
    Aug 2009
    in relation to what?

    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 .

    I would appreciate it if someone could walk me through this, if like me, you find yourself with a lot of free time.

    Let  X be a random variable with probability density function

     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  f(x) = 0 \ \ \,  x \leq 0 (a piecewise function) where  \mu \in R , \sigma > 0 , \varepsilon \in [0,1), \ and \ n \in N are parametres (N and R denote natural and real numbers). Note that when  \varepsilon = 0, f(x) is the pdf corresponding to a long-normal distribution.

    (1) Show that  \int_{- \infty}^{\infty} f(u) du = 1

    (2) Show that the moments  E[X^{\alpha}] for all  \alpha \in \ R do not depend on  \varepsilon or n.

    (3) Show that  M(t) = E[e^{tX}] is infinite for any t > 0

    Good Luck
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  2. #2
    Aug 2009

    first is obvious

    Integral in 1 is just a sum of integral over log-normal density and integral which is reduced (by introducing new variable =( log(x)-mu )/ sigma ) to integral over odd function which equals to zero
    Last edited by kobylkinks; Aug 15th 2009 at 11:04 AM.
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