Let $M_{X,Y}(t_1,t_2) = {t_1^2+t_2^2\over 2}$ be the joint moment generating function of $X,Y$. Is it possible to find the joint density function of $X,Y$?
I´m having a really difficult time in this problem. I don´t know hot to compute the joint density. I would really appreciate any ideas or suggestions.
The MGF is usually defined by $M_(X_1,X_2)(t_1,t_2)=Ee^{t_1X_1+t_2X_2}$. With this definition we have $M_(X_1,X_2)(0,0)=1$ so your function is not a MGF.
I suppose you want to consider $M_(X_1,X_2)(t_1,t_2)=Et_1^{X_1}t_2^{X_2}$. I will answer the question taking this as the definition.
Solution based on guess work: $(X,Y)$ takes just two values $(0,2)$ and$(2,0)$ with probability $\frac 1 2 $ each.