Time as a random variable

Each time measurement is considered as the realization of a random variable

Mean value \(T_t\)=20 mn, standard deviation \(\sigma_1\)=5 mn

  • Density probability function :

    \(\displaystyle f_{T_1}(T) ={1\over \sqrt{2\,\pi} \; \sigma_1} \, \exp \left[{-{ (T-T_t)^2\over 2\,\sigma_1^2}}\right]\)

  • Mean value: \(T_t= \left< T_1\right> = \int_{I\!R} T \; f_{T_1}(T) \, dT\sim {1\over N} \; \sum_{n=1}^N \; T_{1,n}\)

  • Variance: \(\sigma_1^2= \left< T_1^{'2}\right> \)with \(T_1' = T_1 - \left< T_1\right>\)