Consider three rv, X, Y, Z with joint Gaussian distribution. Namely, the first two are a bivariate Gaussian vector, with non zero correlation ρ(x,y) , while Z is independent from the first two. Denoting as usual with µx, µy , µz the means, find the distribution of the sum X + Y + Z
clearly indicating all the properties you use
My doubt is can I calculate the distribution of X+Y, and then do the convolution between Z and what I obtained before?
Yes, if $\ Z\ $ is independent of $\ X\ $ and $\ Y\ $, then $\ X+Y\ $ and $\ Z\ $ are independent, so you can compute the distribution of $\ X+Y+Z\ $ by convolving the distributions of $\ X+Y\ $ and $\ Z\ $.
There's really no need to do that, however, since $\ X+Y+Z\ $ must be normal with mean $\ \mu_X+\mu_Y+\mu_Z\ $ and variance $\ \sigma_X^2+\sigma_Y^2+\sigma_Z^2+$$2\sigma_X\sigma_Y\rho_{XY}\ $, where $\ \sigma_X, \sigma_Y\ $ and $\ \sigma_Z\ $ are the standard deviations of $\ X,Y\ $ and $\ Z\ $, respectively.