多項分布のモーメント母関数の導出

モーメント母関数の定義式 $M_{\mathit{X}}\left(\mathit{\theta }\right)=\sum e^{{\theta }_1{X}_1+{\theta }_2{X}_2+\cdots +{\theta }_n{X}_n}\cdot f\left(x\right)$ より、 $$\begin{array}{rl}{M}_X\left(\mathit{\theta }\right)& =\sum _{n_1+n_2+\cdots +n_k=n}{e}^{{\theta }_1{x}_1+{\theta }_2{x}_2+\cdots {\theta }_k{x}_k}\cdot \frac{n!}{x_1!\cdots x_k!}{p}_1^{x_1}\cdots p_k^{x_k}\\ & =\sum _{n_1+n_2+\cdots +n_k=n}\frac{n!}{x_1!\cdots x_k!}{\left(p_1{e}^{{\theta }_1}\right)}^{x_1}{\left(p_2{e}^{{\theta }_2}\right)}^{x_2}\cdots {\left(p_k{e}^{{\theta }_k}\right)}^{x_k}\end{array}$$ 多項定理 $$\begin{array}{r}{(x_1+x_2+\cdots +x_k)}^n=\sum _{n_1+n_2+\cdots +n_k=n}\frac{n!}{n_1!n_2!\cdots n_k!}{x}_1^{n_1}{x}_2^{n_2}\cdots x_k^{n_k}\end{array}$$ より、 $$\begin{array}{r}{M}_X\left(\mathit{\theta }\right)={\{p_1{e}^{{\theta }_1}+p_2{e}^{{\theta }_2}+\cdots +p_k{e}^{{\theta }_k}\}}^n\end{array}$$ $◼$