I need to use the formula \begin{align*} f_{Y_1}(y_1) = \int_{-\infty}^{\infty} f_{X_{1}X_{2}}(y_1-y_2, y_2)d_{y_2} \end{align*}
to find the pdf of $Y_1 = X_1 + X_2$, where $X_1$ and $X_2$ have the joint pdf $f_{X_{1}X_{2}}(x_1, x_2) = 2e^{-(x_1+x_2)}$ for $0 < x_1 < x_2 < \infty$, zero elsewhere.
First, let $Y_2 = X_2$ so that $X_1 = Y_1-Y_2$ and $X_2 = Y_2$. Then, the bounds $0 < x_1 < x_2 < \infty$ become $0 < y_1-y_2 < y_2 < \infty$, or $y_2 < y_1 < 2y_2 < \infty$. Now using the formula and noting that $0 < y_2 < y_1$, I obtain
\begin{align*} f_{Y_1}(y_1) = 2\int_{0}^{y_1}e^{-y_1}dy_2 = 2y_1e^{-y_1}. \end{align*}
Basically, I am just unsure if my bounds for $y_2$ are correct. Are there any errors in my work? Thank you =)
The correct answer is $2\int_{\frac {y_1} 2}^{y_1} e^{-y_1}dy_2=y_1e^{-y_1}$. [I fact what you have obtained is not density function since it does not integrate to $1$].