Finding density of a sum of two variables 
Let $X$ be exponential r.v with parameter $\lambda$ and $Y$ also
  exponential with parameter $2 \lambda$ and independent of $Y$. Find
  probability density of $X+Y$.

We know 
$$ f_{X+Y}(a) = \int\limits_{-\infty}^{\infty} f_X(x)f_Y(a-x) dx $$
Now, my problem here is to put the right limits of integration. I would say $x > 0$, and hence 
$$ f_{X+Y}(a) = \int\limits_0^{\infty} \lambda e^{- \lambda x} 2 \lambda e^{-2\lambda(a-x)} dx = 2 \lambda^2 \int\limits_0^{\infty}e^{-2 \lambda a} e^{\lambda x }$$
But, then we see that the integral would be divergent. Are my limits of integration wrong?

Integration limits will be based on:
$$ x \ge 0$$ and $$(a-x) \ge 0 $$ Resulting in$$\int_{0}^{a}... $$
