Assume that Y has a beta distribution with parameters a and b. Find the density function of U = 1 - Y.
I know how to do then when they give the density function of Y, but i'm confused here.
First, for the density function of a beta$(\alpha,\beta)$ random variable, see http://en.wikipedia.org/wiki/Beta_distribution.
Now, if $Y$ is beta distributed, then it takes values in $(0,1)$. Hence, $U = 1 - Y$ also takes values in $(0,1)$. In order to find the density of $U$, it is useful to find first its distribution function, and then differentiate it. The distribution function of $U$ at $x \in (0,1)$ is the tail distribution function of $Y$ at $1-x$. By taking complement, you get the distribution function of $U$ expressed in terms of that of $Y$. Differentiating it, you get the density of $U$ expressed in terms of that of $Y$. You should find out that $U$ and $Y$ are very closely related.