E [ X ] = ∫ 0 ∞ t f ( t ) d t {\displaystyle E[X]=\int \limits _{0}^{\infty }tf(t)\,dt}
= ∫ 0 τ t f ( t ) F ( τ ) d t {\displaystyle =\int \limits _{0}^{\tau }{\frac {tf(t)}{F(\tau )}}\,dt}
= ∫ 0 τ t λ e − λ t ( 1 − e − λ τ ) d t {\displaystyle =\int \limits _{0}^{\tau }{\frac {t\lambda e^{-\lambda t}}{(1-e^{-\lambda \tau })}}\,dt}
= 1 ( 1 − e − λ τ ) ∫ 0 τ t λ e − λ t d t {\displaystyle ={\frac {1}{(1-e^{-\lambda \tau })}}\int \limits _{0}^{\tau }{t\lambda e^{-\lambda t}}\,dt}
= 1 ( 1 − e − λ τ ) [ − t e − λ t + − e − λ t λ ] | 0 τ {\displaystyle ={\frac {1}{(1-e^{-\lambda \tau })}}{\Big [}-te^{-\lambda t}+{\frac {-e^{-\lambda t}}{\lambda }}{\Big ]}{\bigg |}_{0}^{\tau }}
= 1 ( 1 − e − λ τ ) [ ( − τ e − λ τ − − e − λ τ λ ) − ( 0 − 1 λ ) ] {\displaystyle ={\frac {1}{(1-e^{-\lambda \tau })}}{\Big [}(-\tau e^{-\lambda \tau }-{\frac {-e^{-\lambda \tau }}{\lambda }})-(0-{\frac {1}{\lambda }}){\Big ]}}
= 1 λ ( 1 − e − λ τ ) [ ( − τ λ e − λ τ + ( 1 − e − λ τ ) ] {\displaystyle ={\frac {1}{\lambda (1-e^{-\lambda \tau })}}{\Big [}(-\tau \lambda e^{-\lambda \tau }+(1-e^{-\lambda \tau }){\Big ]}}
= − τ e − λ τ ( 1 − e − λ τ ) + 1 λ {\displaystyle ={\frac {-\tau e^{-\lambda \tau }}{(1-e^{-\lambda \tau })}}+{\frac {1}{\lambda }}}
E [ X ] = 1 λ − τ ( e λ τ − 1 ) − 1 {\displaystyle E[X]={\frac {1}{\lambda }}-\tau (e^{\lambda \tau }-1)^{-1}}
E [ X 2 ] = ∫ 0 τ t 2 f ( t ) F ( τ ) d t = 2 − λ τ ( λ τ + 2 ) e λ τ − 1 λ 2 {\displaystyle E[X^{2}]=\int \limits _{0}^{\tau }{\frac {t^{2}f(t)}{F(\tau )}}\,dt={\frac {2-{\frac {\lambda \tau (\lambda \tau +2)}{e^{\lambda \tau }-1}}}{\lambda ^{2}}}}
σ 2 = τ e λ τ ( λ τ + e λ τ ( λ τ − 2 ) + 2 ) ( e λ τ − 1 ) 3 {\displaystyle \sigma ^{2}={\frac {\tau e^{\lambda \tau }(\lambda \tau +e^{\lambda \tau }(\lambda \tau -2)+2)}{(e^{\lambda \tau }-1)^{3}}}}
![{\displaystyle E[X]=\int \limits _{0}^{\infty }tf(t)\,dt}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7a9014590b35a68bcc77d073020a9ba2ab6584e0)
![{\displaystyle ={\frac {1}{(1-e^{-\lambda \tau })}}{\Big [}-te^{-\lambda t}+{\frac {-e^{-\lambda t}}{\lambda }}{\Big ]}{\bigg |}_{0}^{\tau }}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a365d227c3312d08e7a268b70ac99a2e3d409570)
![{\displaystyle ={\frac {1}{(1-e^{-\lambda \tau })}}{\Big [}(-\tau e^{-\lambda \tau }-{\frac {-e^{-\lambda \tau }}{\lambda }})-(0-{\frac {1}{\lambda }}){\Big ]}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9c8461ba4600806d961d0e30835d8c579e0efd0e)
![{\displaystyle ={\frac {1}{\lambda (1-e^{-\lambda \tau })}}{\Big [}(-\tau \lambda e^{-\lambda \tau }+(1-e^{-\lambda \tau }){\Big ]}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/20a65ca1dc4ce5b20e85b56230126834dcfe916b)
![{\displaystyle E[X]={\frac {1}{\lambda }}-\tau (e^{\lambda \tau }-1)^{-1}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/344e2156ae635d9920fcd82e2a34a9c7e3850135)
![{\displaystyle E[X^{2}]=\int \limits _{0}^{\tau }{\frac {t^{2}f(t)}{F(\tau )}}\,dt={\frac {2-{\frac {\lambda \tau (\lambda \tau +2)}{e^{\lambda \tau }-1}}}{\lambda ^{2}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f08af14563947dc59f06d48df31e9b880f3014c3)
