
Density Functions
Let X be uniform on (pi/2, pi/2) and let Y = atan(X), a > 0. Find $\displaystyle f_Y(y)$ in terms of $\displaystyle f_X$.
I know how to do this, but somehow I'm not getting the answer that the book has. The book says the answer is $\displaystyle \frac{a}{\pi(a^2 + y^2)}$ but I keep getting $\displaystyle \frac{a^2}{\pi(a^2 + y^2)}$.
I think maybe my problem is with the indicator function $\displaystyle 1_{(\frac{\pi}{2},\frac{\pi}{2})}(tan^{1}(\frac{y}{a}))$. 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.

Hello,
Let $\displaystyle t=a \tan(x)$
then $\displaystyle x=\arctan(x/a)$
differentiate (don't forget the chain rule ! I'm sure this is where you made your mistake) :
$\displaystyle dx=\frac{dt/a}{1+(t/a)^2}=\frac{a ~ dt}{a^2+t^2}$
which gives the result the book finds.

Oh wow, duh! Thanks a lot :)