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 $M_\boldsymbol{X} \left(\boldsymbol{\theta}\right)=\sum{e^{\theta_1X_1+\theta_2X_2+ \cdots +\theta_nX_n} \cdot f \left(x\right)}$ より、 \begin{align} M_X \left(\boldsymbol{\theta}\right)&=\sum_{n_1+n_2+ \cdots +n_k=n}{e^{\theta_1x_1+\theta_2x_2+ \cdots \theta_kx_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_1e^{\theta_1}\right)^{x_1} \left(p_2e^{\theta_2}\right)^{x_2} \cdots \left(p_ke^{\theta_k}\right)^{x_k}}\\ \end{align} 多項定理 \begin{align} \left(x_1+x_2+ \cdots +x_k\right)^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{align} より、 \begin{align} M_X \left(\boldsymbol{\theta}\right)= \left\{p_1e^{\theta_1}+p_2e^{\theta_2}+ \cdots +p_ke^{\theta_k}\right\}^n \end{align} $\blacksquare$