I would like to find
\begin{align*} \mathbb{E}[ q | q + v = k] \\ \mbox{where} \quad q \sim \mbox{exp}(\lambda) \\ \mbox{and} \quad v \sim \mbox{exp}(\mu) \\ \mbox{and} \quad k = \mbox{some constant} \end{align*}
I have tried the following approach:
\begin{align*} \mathbb{E}[ q | q + v = k] = \int\limits_0^{\infty} \frac{f_{q \cap (q + v)}(q)}{f_{q+v}(k)} \mbox{d}q \end{align*}
For which I have already found the convolution density of $f_{q+v}$ but I don't know how to find the density $f_{q\cap(q+v)}$. Moreover, I am unsure if this approach is correct and / or efficient in general.
In addition to answer suggestions, I would greatly appreciate if someone could reference a textbook that deals well with issues such as this.