I am having difficulty with one equality in the following proof: Note that $M_Y(t)$ is the moment generating function of the random variable $Y$. Not sure if this notation is universal.
Why is it true that $M_Y(t)= \prod_{i=1}^{n}M_{X_i}(c_it)$ . From what I understand $M_Y(t)=e^{t(c_1X_1+\cdots +c_nX_n)}f(Y)$ where $f(Y)$ is the normal probability density function associated to Y and I am not sure how this equality follows.
Since $X_1, \ldots, X_n$ are independent, so are $\exp(t c_1 X_1), \ldots, \exp(t c_n X_n)$, so $$\eqalign{M_Y(t) &= \mathbb E\left[ e^{t(c_1 X_1 + \ldots + c_n X_n)}\right] = \mathbb E\left[ e^{tc_1 X_1} \ldots e^{t c_n X_n}\right] = \mathbb E\left[e^{tc_1 X_1}\right] \ldots \mathbb E\left[e^{tc_n X_n}\right]\cr &= M_{X_1}(tc_1) \ldots M_{X_n}(t c_n)}$$