# Density Functions

• Nov 11th 2009, 12:55 PM
azdang
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.
• Nov 11th 2009, 02:58 PM
Moo
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.
• Nov 11th 2009, 03:02 PM
azdang
Oh wow, duh! Thanks a lot :)