Given integrable functions f and g on $\mathbb{R}$ define $f*g$, the convolution of $f$ and $g$ by
$f*g(x) = \int_{-\infty}^\infty f(y)g(x-y) dy$, So i have to show that $M_Z(t) = M_X(T)M_Y(t)$ which is pretty simple. But how do i use this to deduce that $X+Y$ has density $f*g$ ?
I then have to go on and calculate the density of $X+Y$ where X and Y are independent standard normal random variables. For this i have
$f_Z = \int_{-\infty}^\infty f_{X,Y}(y,x-y) dy = \int_{-\infty}^\infty f_{X}(y) f_Y(x-y) dy = (1/2\pi)\int_{-\infty}^\infty e^{-y^2/2}e^{-(x-y)^2/2}dy$
Is this the correct integral?