I would like to find the conditional expectation of a random variable $q$ which is an exponential random variable with $\mbox{pdf}(q) = \lambda e^{-\lambda q}$ conditional on $q + v > k$, where $k$ is a constant and $v \sim \mbox{exp($\nu$)}$, ie another, independent exponential random variable with parameter $\nu$.
I am trying something like $\mathbb{E}[q | q + v > k] = \frac{\int\limits_0^{\infty}\mbox{``density'' of}\quad (q \quad \cap \quad q + v > k) \mathbb{d}q}{\mathbb{P}[q + v > k]}$ but cannot get this to work
Using memorylessness of an exponential: $E(q|q>t)=E(q)+\max(t,0)$ for fixed $t$ and conditioning on $v$ we get:
$$E(q|q+v>k)=E(q|q>k-v)=E(q)+E(\max(k-v,0))=E(q)+E(k-v;k-v>0)=E(q)+E(k-v;v<k)=\frac 1 {\lambda_1}-\int_0^k(k-v)\,de^{-\lambda_2v}=\frac 1 {\lambda_1}+k-\int_0^ke^{-\lambda_2 v}\,dv=\frac 1 {\lambda_1}+k-\frac {1-e^{\lambda_2k}}{\lambda_2}.$$ Similarly we get $$E(v|q+v>k)=\frac 1 {\lambda_2}+k-\frac {1-e^{-\lambda_1k}}{\lambda_1}$$ And adding them up yields: $$E(q+v|q+v>k)=2k+\frac {e^{-\lambda_1k}}{\lambda_1}+\frac {e^{-\lambda_2k}}{\lambda_2}$$