Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / M / 1 / b / iii / Solution

Take $Y=X_1+X_1^{\prime }$, where $X_1^{\prime }$ is an independent copy of $X_1$. The preceding factorization gives $\mathbb{E}{e}^{tY}=M_1(t{)}^2$. Using $2X_1$ would instead give $M_1(2t)$, so an independent copy is required.