指数分布
定义
指数分布的期望
\[EX = \frac{1}{\lambda}
\]
证明
\[EX = \int_{-\infty}^{+\infty}xf(x)dx = \int_{0}^{+\infty}x\lambda e^{-\lambda x}dx = -\int_{-0}^{+\infty}xde^{-\lambda x} = \int_{0}^{+\infty}e^{-\lambda x}dx = \frac{1}{\lambda }
\]
指数分布的方差
\[DX = \frac{1}{\lambda ^ 2}
\]
证明
\[EX^{2} = \int_{-\infty }^{+\infty }g(x)f(x)dx = \int_{0}^{+\infty }x^{2}\lambda e^{-\lambda x}dx = -\int_{0}^{+\infty }x^{2}de^{-\lambda x}
\]
\[= -x^2e^{-\lambda x}|_0^{+\infty} + 2\int_{0}^{+\infty}xe^{-\lambda x}dx = \frac{2}{\lambda }\int_{0}^{+\infty}e^{-\lambda x}dx = \frac{2}{\lambda ^{2}},
\]
所以，
\[DX = EX^{2} - (EX)^{2} = \frac{2}{\lambda ^{2}} - (\frac{1}{\lambda})^{2} = \frac{1}{\lambda ^{2}}.
\]
