Solution
= Solution
Let $X_1'$ be an independent copy of $X_1$ and take $Y=X_1+X_1'$. Then
$$
\boxed{M_Y(t)=M_1(t)^2.}
$$
<Independence> of the copy is essential. The variable $2X_1$ would instead have <moment-generating function> $M_1(2t)$.
= Solution
Let $X_1'$ be an independent copy of $X_1$ and take $Y=X_1+X_1'$. Then
$$
\boxed{M_Y(t)=M_1(t)^2.}
$$
<Independence> of the copy is essential. The variable $2X_1$ would instead have <moment-generating function> $M_1(2t)$.