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.
Best Answer
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 $1$.