PDF of sum of random variables (with uniform distribution) How can I solve this:
Random variables $X,Y$ ~ Unif$(0, 1)$ are independent. Calculate the probability density function of sum $X + 3Y$.
I couldn't find a sum for uniformally  distributed random variables. I assume I have to go straight to the PDF and solve it that way.

One way to avoid explicit convolution (although convolution is always involved, in the end), is to define $Z = 3Y \sim \text{Uniform}(0, 3)$, and look at the distribution of $(X, Z)$ in the $x$-$z$ plane: a uniformly distributed rectangle.
Within this rectangle, the bands of equal values of $X+Z$ correspond to diagonal stripes.  The lengths of these stripes, where $X+Z =$ some value $w$, is proportional to the value of the PDF $f_{X+Z}(w)$.  All you need to do, then, is to find the proportionality constant that makes it a PDF; that is, it must integrate to $1$.
