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Derive the moment generating function for the Irwin-Hall distribution - the distribution of the sum of $n$ independent and identically distributed random variables, each with standard uniform distribution $U[0, 1]$

This question has really stumped me as im not a big fan of Irwin-Hall distributions so any help will be appreciated.

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$\{U_i\}_{i=1}^n$ n iid uniform distributed variables, and $Z=\sum_{i=1}^{n}{U_i}$

The moment generating function of $Z$ is $$\phi_Z(u)=E(e^{uZ})$$

$$\phi_Z(u)=E(e^{u\sum_{i=1}^{n}{U_i}})$$ $$=E(\prod_{i=1}^{n}e^{u{U_i}})$$

By independence , we have $$E(\prod_{i=1}^{n}e^{u{U_i}})=\prod_{i=1}^{n}E(e^{u{U_i}})$$ Because the $U_i$'s have the same distribution , we have

$$\phi_Z(u)=E(e^{u{U_1}})^n$$ or $$\phi_Z(u)=\phi_{U_1}(u)^n$$ where $\phi_{U_1}$ is the moment generating function of a uniform variable, which is defined by

$$\phi_{U_1}(u)=\frac{e^u-1}{u}$$ if $u$ non nil, $1$ otherwise

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