指数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 $M_X\left(\theta \right)={\int }_{x=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{\theta x}\cdot f\left(x\right)dx$ より、 $$\begin{array}{rl}{M}_X\left(\theta \right)& ={\int }_{-\mathrm{\infty }}^0{e}^{\theta x}\cdot 0\cdot dx+{\int }_0^{\mathrm{\infty }}{e}^{\theta x}\cdot \lambda e^{-\lambda x}dx\\ & =\lambda {\int }_0^{\mathrm{\infty }}{e}^{\theta x-\lambda x}dx\\ & =\lambda {\int }_0^{\mathrm{\infty }}{e}^{-(\lambda -\theta )x}dx\\ & =-\lambda \cdot \frac{1}{\lambda -\theta }{\left[e^{-(\lambda -\theta )x}\right]}_0^{\mathrm{\infty }}\\ & =-\lambda \cdot \frac{1}{\lambda -\theta }(\underset{x\to \mathrm{\infty }}{lim}{e}^{-(\lambda -\theta )x}-e^{-0})\\ & =\frac{\lambda }{\lambda -\theta }\end{array}$$ $◼$