Finding density of a sum of two variables

Question

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?

Answer 1

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

Answer 2

Note that the exponential RV can only take positive values. Hence, your limits should ensure that $$X,Y>0$$

As such, $$a-x>0\implies x<a$$

and from above, $$x>0$$

Thus the limits should be from $0$ to $a$.