Let $X$ and $Y$ be two independent continuous random variables each uniformly distributed between $[0,1]$ (i.e., $f_x (a)=f_y (a)=1,0\leq a \leq 1$). Please find the density function of the sum of the random variables $X$ and $2Y$, i.e., $f_{X+2Y} (a)$.
Am I doing this right? :[ As you can see, I get a little confused at the bottom, and I'm just kind of stuck.
Any help would be greatly appreciated, thank you.
My work. Cumulative distribution function of $X+ 2Y$: $ \begin{align*} F_{X+2Y} (a) &= P(X+2Y \leq a) \\ &= \iint_{x+ y \leq a} f_X(x) f_Y(y) ~ dy ~dx \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{a - 2y} f_X(x) f_Y(y) ~ dx ~dy \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{a - 2y} f_X(x) ~ dx\ f_Y(y) ~dy \\ &= \int_{-\infty}^{\infty} F_X(a-2y) f_Y(y) ~dy \end{align*} $
The probability density function: $ \begin{align*} f_{X+2Y}(a) &= \frac{d}{da} \int_{-\infty}^{+\infty} F_X(a-2y) f_Y(y) ~ dy \\ &= \int_{-\infty}^{+\infty} \frac{d}{da} F_X(a-2y) f_Y(y) ~ dy \\ &= \int_{-\infty}^{+\infty} f_X(a-2y) f_Y(y) ~ dy \end{align*} $
For $[0,1]$: $ f_X(a) = f_Y(a) = \begin{cases} 1, & 0 \lt a \lt 1, \\ 0, &\text{otherwise}. \end{cases} $
$ f_{X + 2Y}(a) = \int_{-\infty}^{+\infty} f_X(a-2y) f_Y(y) ~ dy \quad \text{for } 0 \lt a \lt 1. $
$ f_{X + 2Y}(a-2) = \int_0^a dy = a. $