Distribution of two uniform variables

Question

Let $X_1\sim U(0,10)$ and $X_2 \sim U(0, 30)$. Find the distribution of $S=\frac{1}{2}(X_1+X_2)+3$.

I tried to find the distribution for $S=X_1+X_2$ using convolution ($f_S(s)=\int f_{X_1}(s-X_2)f_{X_2}(X_2)dX_2$):

$f_S(s)=\left\{\begin{array}{rcl}\frac{1}{300}s&\text{if}&0\leq s\lt10,\\\frac{1}{30}&\text{if}&10\leq s \lt 30,\\\frac{2}{15}-\frac{1}{300}s&\text{if}&30 \leq s\leq 40,\\0&&\text{otherwise}\end{array}\right.$

However, I am having trouble trying to find the distribution for $S=\frac{1}{2}(X_1+X_2)+3$.

Answer 1

Hint: If $Z=\frac 1 2Y+3$ then $f_Z(z)=2f_Y(2(z-3))$.