Let $X$ and $Y$ be independent non-negative random variables, with densities $f$ and $g$ respectively. Find the joint density of $U=X$ and $V=X+a Y$, where $a$ is a positive constant.

Let $X$ and $Y$ be independent and exponentially distributed random variables, each with density

$$f(x)=\lambda e^{-\lambda x}, \quad x \geqslant 0$$

Find the density of $X+\frac{1}{2} Y$. Is it the same as the density of the random variable $\max (X, Y) ?$