離散型確率変数(discrete random variable) の一様分布(uniform distribution)の積率母凾数(moment-generating function)

$$f_X(x)=\{\begin{array}{ll}{\displaystyle \frac{1}{n}}& x\in \{1,2,\dots ,n\}\\ {\displaystyle 0}& x\notin \{1,2,\dots ,n\}\end{array}$$ $$\begin{array}{rcl}{\displaystyle M_X(t)}& \equiv & {\displaystyle E[{\mathrm{e}}^{tX}]}\\ & =& {\displaystyle \sum _{x=1}^n{\mathrm{e}}^{tx}\left(\frac{1}{n}\right)}\\ & =& {\displaystyle \frac{1}{n}\sum _{x=1}^n{\mathrm{e}}^{tx}}\\ & =& {\displaystyle \frac{1}{n}({\mathrm{e}}^t+{\mathrm{e}}^{2t}+\cdots +{\mathrm{e}}^{nt})}\\ & =& {\displaystyle \frac{1}{n}\frac{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{{\mathrm{e}}^t-1}\dots a+ar+a{r}^2+\cdots +a{r}^{n-1}=\frac{a(1-r^n)}{1-r}=\frac{a(r^n-1)}{r-1}(r\ne 1)}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{n({\mathrm{e}}^t-1)}}\end{array}$$