Let $Z=X+Y$ where $X$~$N(\mu,\sigma^2)$ and $Y$~$N(0,1)$ are independents. Find the joint density of Z and X.
This is the first time I see something like that, look what I did below:
I know that $Z=X+Y$~$N(\mu,\sigma^2+1)$, but it does not help a lot, so I developed through Jacobian
$Z=X+Y$ and $W=X$, this I have that Jacobian $J=-1\Rightarrow |J|^{-1}=1$, and solving that equations $X=W$ and $Y=Z-W$, then
$$f_{x,y}(x,y)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}*\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}y^2}$$ $$f_{x,y}(x,y)=\frac{1}{2\pi\sigma}e^{-\frac{1}{2\sigma^2}[(x-\mu)^2+\sigma^2y^2]}=\frac{1}{2\pi\sigma}e^{-\frac{1}{2\sigma^2}[x^2-2\mu x+\mu^2+\sigma^2y^2]}$$ replacing $X$ and $Y$ $$f_{z,w}(z,w)=\frac{1}{2\pi\sigma}e^{-\frac{1}{2\sigma^2}[w^2-2\mu w+\mu^2+\sigma^2(z-w)^2]}=\frac{1}{2\pi\sigma}e^{-\frac{1}{2\sigma^2}[w^2-2\mu w+\mu^2+\sigma^2(z^2-2zw+w^2)]}$$
Now I'm stuck, I can not simplify anything and do not even know you have to do something.
Since $Z\sim N(\mu,1+\sigma^2)$ and $X\sim(\mu,\sigma^2)$ are jointly normal, it suffices to know their covariance.
$$ E[ZX]-E[Z]E[X] = E[(X+Y)X] - \mu^2 = E[X^2] + E[XY] - \mu^2 = (\sigma^2 + \mu^2) + 0 - \mu^2 = \sigma^2 $$
Thus $(Z,X)$ is a bivariate normal with mean $$ \left(\begin{array}{c} \mu\\ \mu \end{array}\right) $$ and covariance matrix $$ \left(\begin{array}{cc} \sigma^2+1 & \sigma^2\\ \sigma^2 & \sigma^2 \end{array} \right) $$