Solution
= Solution
Take $Y=aX_1+b$. Directly,
$$
\boxed{\mathbb E(e^{tY})=e^{bt}\mathbb E(e^{atX_1})=e^{bt}M_1(at).}
$$
The formula holds wherever the <moment-generating function> on the right is finite.
= Solution
Take $Y=aX_1+b$. Directly,
$$
\boxed{\mathbb E(e^{tY})=e^{bt}\mathbb E(e^{atX_1})=e^{bt}M_1(at).}
$$
The formula holds wherever the <moment-generating function> on the right is finite.