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Math Help - Density Functions

  1. #1
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    Density Functions

    Let X be uniform on (-pi/2, pi/2) and let Y = atan(X), a > 0. Find f_Y(y) in terms of f_X.

    I know how to do this, but somehow I'm not getting the answer that the book has. The book says the answer is \frac{a}{\pi(a^2 + y^2)} but I keep getting \frac{a^2}{\pi(a^2 + y^2)}.

    I think maybe my problem is with the indicator function 1_{(\frac{-\pi}{2},\frac{\pi}{2})}(tan^{-1}(\frac{y}{a})). I thought this would be equal to 1, but is it possible it is equal to 1/a? Then, the answer given in the book would make sense.
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  2. #2
    Moo
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    Hello,

    Let t=a \tan(x)
    then x=\arctan(x/a)

    differentiate (don't forget the chain rule ! I'm sure this is where you made your mistake) :

    dx=\frac{dt/a}{1+(t/a)^2}=\frac{a ~ dt}{a^2+t^2}

    which gives the result the book finds.
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  3. #3
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    Oh wow, duh! Thanks a lot
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