Moment generating function of i.i.d

Question

I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf

I have two questions

1. I know about moment generating function of say X1, then how was the formula for the iid sequence X1...Xn derived?
2. How did the author do

$$\Pi_{i=1}^nEe^{t_iX_i}=\Pi_{i=1}^ne^{\frac{t_i^2}{2}}?$$

Answer 1

1. Let $X_i$ have cdf $F_X(x_i)$, so its mgf is $\int_{\Bbb R}\exp t_iX_i dF_X(x_i)$. By independence, the mgf of $\sum_i X_i$ is $\int_{\Bbb R^n}\exp\sum_i t_iX_i \prod_idF_X(x_i)=\prod_i\int_{\Bbb R}\exp t_iX_i dF_X(x_i)$. In other words, we just multiply the characteristic functions.
2. The claimed result follows exactly if each $X_i\sim N(0,\,1)$, as then $E\exp t_iX_i=\exp t_i^2/2$. (For more general distributions of mean $0$ and variance $1$, the central limit theorem implies the result is approximately valid.) Indeed, the substitution $y=x-t$ gives $$\int_{\Bbb R}\frac{1}{\sqrt{2\pi}}\exp (tx-\tfrac{x^2}{2})dx=\int_{\Bbb R}\frac{1}{\sqrt{2\pi}}\exp\tfrac{t^2-y^2}{2}dy=\exp\tfrac{t^2}{2}.$$