Say X is a uniform random var on [-1,1], so$\displaystyle f_X(x)=1/2$ on [-1,1]. I want to find the density of $\displaystyle Y=g(X)=X^3$ over [g(-1),g(1)]=[-1,1]. g is a monotonic function so everything should be kosher. I look up the change of variable formula on wikipedia:

$\displaystyle f_Y(y)=|1/g'(g^{-1}(y))|f_X(g^{-1}(y))$

So I do some algebra:

$\displaystyle g(x)=x^3 $

$\displaystyle g'(x)=3x^2 $

$\displaystyle g^{-1}(y)=y^{1/3} $

so $\displaystyle g'(g^{-1}(y))=g'(y^{1/3})=3y^{2/3}$

thus

$\displaystyle f_Y(y)=|1/3y^{2/3}|*(1/2)$

But there's a singularity/asymtopes at y=0!... am I doing something wrong? The books I have conveniently choose random vars so there is no divide by zero (for example, if X was on [1,2] or some other interval where g(x) doesn't cross the x-axis, everything works fine).