(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

IfXis 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 pdffof_{Z}(z)Z= arctan(x).

3. The attempt at a solution

IfZ=g(X), theng(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?

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# Homework Help: Functions of Random Variables

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