$(X,Y)$ random vector with characteristic function
$\phi_{X,Y}(s,t)=e^{-\frac{(s+t)^2+t^2}{2}}(1+ie^{-\frac{1}{2}}sinh(s+t))$.
a) Find $EX$ if it exists.
b) Justify of which type $X+Y$ is distribution and find this distribution.
a) I first found $\phi_X(s)$ by putting $0$ in place of $t$: $\phi_X(s)=\phi_{X,Y}(s,0)$.
Then I checked that $\phi_X''(0)< \infty$ - there is a theorem that $\phi_X^{(2n)}(0)< \infty \implies EX^{2n}<\infty$.
So $EX$ exists and is equal to $\phi_X'(0)$.
Is it okay?
b) Do I first have to find the distribution of $(X,Y)$ using inversion formula and calculate the double integral of $\phi$ or is there a simpler way?