Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$.
What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$?
What I did.
\begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx dy \end{align*}
For the case p=1. I can show that $E[|aX-bY|]=\frac{b^2}{3a}+\frac{a-b}{2}$ if $b \le a$ and $E[|aX-bY|]=\frac{a^2}{3b}+\frac{b-a}{2}$. For, the case $p>1$ the intgration becomes difficult (at least for me). I also think an lower bound on it will suffice for my purposes.
Is this an way to solve this? Also, is this something that is commonly encountered in probability?