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Math Help - Density function ln(x)

  1. #1
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    If ln(x) has a normal distribution N(0,1) how do I find and prove the distribution for x?

    And if S(t)=S(0)exp((mu-(sigma^2)/2)t+sigma*sqrt(tao)*G), where G N(0,1) what is the density function for S(t)?
    Last edited by CaptainBlack; November 4th 2009 at 02:17 PM.
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  2. #2
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    How do you guys do to make those nice equations?
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    Hello,
    Quote Originally Posted by memanuelson View Post
    And if S(t)=S(0)exp((mu-(sigma^2)/2)t+sigma*sqrt(tao)*G), where G N(0,1) what is the density function for S(t)?
    Click on the "picture" to see the code
    Anyway, there's a latex sub-forum, where you may be able to learn some things about writing equations.

    S(t)=S(0) \exp\left(\frac{\mu-\sigma^2}{2} \cdot t+\sigma \sqrt{\tau} G\right)

    But what is "tao" ?

    If ln(x) has a normal distribution N(0,1) how do I find and prove the distribution for x?
    The pdf of T=\ln(x) is \frac{1}{\sqrt{2\pi}} \cdot  e^{-\frac{t^2}{2}}

    Now, I can never remember the method you guys use... But mine is similar, using the Jacobian of the transformation.

    For any bounded and continuous function f, we have :
    \mathbb{E}(f(X))=\mathbb{E}(f(e^T))=\int_{-\infty}^\infty f(e^t) \cdot \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{t^2}{2}} ~dt

    Make the substitution x=e^t :
    dx=e^t ~dt=x dt

    So \mathbb{E}(f(X))=\int_0^\infty f(x) \cdot \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{(\ln(x))^2}{2}} \cdot \frac 1x ~dx

    Thus the pdf of X is \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{(\ln(x))^2}{2}} \cdot \frac 1x, for x>0
    Which is the pdf of the log-normal distribution, with parameters (0,1).
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    Thank you so much!
    Last edited by memanuelson; May 4th 2009 at 11:45 AM.
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