I have this problem on a past exam where I'm trying to find the convolution of $X\sim N(1,2)$ and $Y$ where $Y$ has pdf $P(Y=-1)=P(Y=1) = 1/2$. However I didn't get a particularly confidence instilling result.
My solution relied on asserting the following equality:
$$f(z) = \int_\mathbb{R} f(z-x) (\frac{1}{2} \delta_{-1} + \frac{1}{2}\delta_{1}) dx = \frac{1}{2} \phi(z+1|1,2) + \frac{1}{2} \phi(z-1|1,2)$$
and then from this you get the result that we have a mixture of 2 normal random variables. Is this correct though? I was thinking that the result would be tidier, because this looks a bit messy.
Thanks for the help!
$$\begin{align} f_Z(z) &= \phi(z|1,2)\star\left( \frac{1}{2} \delta(z-1) + \frac{1}{2}\delta(z+1) \right)\\ & = \frac{1}{2} \phi(z-1|1,2) + \frac{1}{2} \phi(z+1|1,2)\\ & = \frac{1}{2} \phi(z|0,2) + \frac{1}{2} \phi(z|2,2)\end{align}$$