moment generating function , for iid

Question

since this pdf is continuous between 0 and 1 i was wondering how i would find the moment generating function of the sum of X(i) .

Answer 1

$$M_{Y}(t)=E[e^{tY}]=E[e^{t\sum_{i=1}^{n}{X_i}}]=E[\prod_{i=1}^{n}e^{t{X_i}}]$$

Using the independence of the $X_i$'s

$$E[\prod_{i=1}^{n}e^{t{X_i}}]=\prod_{i=1}^{n}E[e^{t{X_i}}]$$

Using the density function of $X_i$

$$E[e^{t{X_i}}]=\int_{0}^{1}{e^{t{x}}2xdx}=\frac{2(e^t(t-1)+1)}{t^2}$$

Finally,

$$M_{Y}(t)=\left(\frac{2(e^t(t-1)+1)}{t^2}\right)^n$$