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Homework Help: Functions of Random Variables

  1. Oct 9, 2007 #1

    cepheid

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    1. The problem statement, all variables and given/known data

    If X is represented by the Gaussian distribution, that is,

    [tex] f_{X}(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{x^2}{2\sigma^2})} [/tex]

    find an expression for the pdf fZ(z) of Z = arctan(x).

    3. The attempt at a solution

    If Z =g(X), then g(X) is multivalued unless the range of the function is restricted to [itex] Z \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) [/itex]

    Under this condition, the function is one to one, and so the probability that z will be in some interval (z, z + dz) is equal to the probability that x will be in the corresponding interval (x, x + dx). In other words,

    [tex] |f_Z(z) dz| = |f_X(x) dx| [/tex]

    [tex] f_Z(z) = \left| \frac{dx}{dz} \right| f_X(x) [/tex]

    [tex] = \frac{d}{dz} (\tan z) f_X(x) [/tex]

    [tex] = (\sec^2(z)) f_X(x) [/tex]

    [tex] = (1+\tan^2(z)) \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{\tan^2(z)}{2\sigma^2})} [/tex] ​

    Am I doing this right?
     
    Last edited: Oct 9, 2007
  2. jcsd
  3. Oct 10, 2007 #2

    EnumaElish

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    I don't see a problem; but you'll need to verify that it integrates to one between the bounds.
     
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