Transformation of a random variable bijectivity How does one determine the density of a transformed random variable?
I understand the theorem, but I have a really hard time applying it. For example, if the transformation is $t(X) = X(1-X) = X - X^2$, and $X$ is uniformly distributed on $(0,1)$. $t$ is not bijective from $(0,1)$ to anywhere, so what does one do? 
I can split it up into two, but then I need to find the inverse of a "split" function. How do I do that?
 A: The standard treatment for calculating the density of a transformed random variable expects the transformation to be one-to-one. If the transformation is not one-to-one, blindly plugging into the formula will often get you an answer that is off by a constant factor that's hard to track down.
Often your best bet is to look for some kind of symmetry that allows you to reduce to the one-to-one case. In your example: for the transformation $t=g(x):=x-x^2=x(1-x)$, note that $g$ is symmetric about $x=1/2$ (in fact $g$ is a parabola with vertex at $(x,g(x))=(1/2,1/4)$). Moreover, the random variable $X$ has a distribution which is also symmetric about $x=1/2$. So the density of $t(X)$ would be the same if $X$ were uniform over the interval $(0,1/2)$, in which case $g(x)$ would be invertible. 
If you apply the change of variables formula to the original problem, the density will be off by a factor of two compared to what you get using alternative approaches (e.g. by calculating $P(X-X^2\le t)$ and differentiating wrt $t$). Note that restricting $X$ to the interval $(0,1/2)$ forces the density of $X$ to be twice as high, which explains that factor of 2.
