If X and Y are 2 independent random variables with Exponential($\lambda$) distributions, my understanding is that their convolution (Z = X + Y) is given by:
$f_Z(z) = -\lambda ze^{-\lambda z}$
The convolution formula is supposed to be:
$f_Z(z) = \int_0^z f_X(z-w)f_Y(w) dy$
$ = \int_0^z \lambda e^{-\lambda (z-w)} \lambda e^{-\lambda (w)} dy$
$ = \int_0^z \lambda^2 e^{-\lambda z} dy$
$ = \lambda^2 y e^{-\lambda z}$ evaluated on $[0, z]$
$ = \lambda^2 z e^{-\lambda z}$
This is supposed to be equivalent to the Gamma Distribution given by: $$\frac{\lambda^2 z^{\alpha-1}}{(\alpha - 1) !}e^{-\lambda z}$$ with $\alpha = 2$
What happened to the -ve sign in $-\lambda$? Nothing, I made a mistake in the integration.
