I am trying to derive the distribution of $Z = X + Y$, where $X$ and $Y$ are normally distributed and but not necessarily independent random variables.
To make the math easier, I started by standardizing $X$ and $Y$, ie created variables $\hat{X}=\frac{X-\mu_X}{\sigma_X}$ and $\hat{Y}=\frac{Y-\mu_Y}{\sigma_Y}$, and then defined $\hat{Z}=\hat{X}+\hat{Y}$.
From here I derived that $F_{\hat{Z}}(\hat{z})=-\frac{1}{\sqrt{2\pi(2+2\rho})}exp(-\frac{\hat{Z}^2}{2(2+2\rho)})$ where $\rho$ is the correlation between $\hat{X}$ and $\hat{Y}$.
But now I've realized that I'm not sure if it is allowed take the distribution $F_{\hat{X}+\hat{Y}}(\hat{X}+\hat{Y})=-\frac{1}{\sqrt{2\pi(2+2\rho})}exp(-\frac{(\hat{X}+\hat{Y})^2}{2(2+2\rho)})$ and then do another transformation to $X=\hat{X}\sigma_X+\mu_X$, $Y=\hat{Y}\sigma_Y+\mu_Y$ to obtain the distribution of $Z$, or if I need to treat $\hat{Z}$ as a single random variable in which case I don't think it would be possible to transform from $\hat{Z}$ to $Z$, and I would have to instead just start with $Z=X+Y$ rather than with the standardized normal variables.