Let $X_1$ and $X_2$ be independent standard normal random variables. Let $Y_1=X_1+X_2$ and $Y_2=X_1^2+X_2^2$. Show that the joint moment generating function of $Y_1$ and $Y_2$ is $\frac{1}{1-2 t_2} \exp \left(\frac{t_1^2}{1-2 t_2}\right)$ for $-\infty<t_1<\infty$ and $-\infty<t_2<\frac{1}{2}$
My answer is: \begin{aligned} & X_1, X_2 \sim N(0,1), Y_1=X_1+X_2, Y_2=X_1^2+X_2^2 \rrbracket \\ & E e^{t 1 Y_1+t 2 Y_2}=E e^{t 1\left(X_1+X_2\right)+t 2\left(X_1^2+X_2^2\right)} \rrbracket \\ & X_1+X_2 \sim(0,2) \rrbracket \\ & E e^{t 1\left(X_1+X 2\right)}=e^{\mu t+\frac{\sigma^2 t 1^2}{2}}=e^{t 1^2} \rrbracket \\ & X_1^2+X_2^2 \sim \operatorname{chisq} \cdot X_2^2, m g f:\left(\frac{1}{1-2 t 2}\right)^{\frac{p}{2}}, E e^{t 2\left(X_1^2+X_2^2\right)}=\frac{1}{1-2 t 2} \rrbracket \\ & E e^{t 1 Y_1+t 2 Y_2}=e^{t 1^2}\left(\frac{1}{1-2 t_2}\right) \end{aligned}
I don't know why it is incorrect