Assuming that $X_1$ ~ $exponential( \lambda _1) \space and \space X_2$ ~ $ exponential( \lambda _2)$ are independent, use the cumulative distribution function technique to find the cumulative distribution function of the sample mean $X=(X_1 +X_2)/2$.
My book uses what it terms as the "cumulative distribution function technique" as a way to express $F_y (y)$ in terms of $F_x(x)$, yet is really unclear how to do so. What my initial thoughts are that I must write the pdf of $Y$ as $F_Y(y)=P(Y \leq y) = P(g(X) \leq y)$ and then algebraically transform $F_Y (y)$ into $F_X(x)$. Is this the gist of it, if anyone is familiar with this "technique?"
Yes, that is the CDF technique. $F_Y(y)~{= \mathsf P((X_1+X_2)/2\leq y) \\ =\int_0^{2y} f_{X_1}(x)\,F_{X_2}(2y-x)\operatorname d x}$