Forskjell mellom versjoner av «Combining measurements of time periods with unequal number of periods»
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Linje 11: | Linje 11: | ||
by setting the first derivative to zero: | by setting the first derivative to zero: | ||
− | <math>{\partial\chi^2\over {\partial \bar{T}}} = 0 = | + | <math>{\partial\chi^2\over {\partial \bar{T}}} = 0 = \sum^N_{i=1}{-2n_i(t_i-n_i\bar{t})\over \sigma_t^2}.</math> |
− | We solve for the mean period: <math>\bar{T}={\sum^N_{i=1}n_it_i\over \sum^N_{i=1}n_i^2}</math> | + | We solve for the mean period: <math>\bar{T}={\sum^N_{i=1}n_it_i\over \sum^N_{i=1}n_i^2}.</math> |
To find the uncertainty on <math>\bar{T}</math> we use the second derivative of the <math>\chi^2</math>: | To find the uncertainty on <math>\bar{T}</math> we use the second derivative of the <math>\chi^2</math>: | ||
<math>{1\over \sigma^2} = {1\over 2}{\partial^2\chi^2\over\partial\bar{T}^2} = | <math>{1\over \sigma^2} = {1\over 2}{\partial^2\chi^2\over\partial\bar{T}^2} = | ||
− | \sum^N_{i=1}{n_i^2\over\sigma_t^2}</math> | + | \sum^N_{i=1}{n_i^2\over\sigma_t^2},</math> |
which gives us | which gives us |
Revisjonen fra 4. feb. 2013 kl. 18:17
Suppose you want to estimate the mean period N measurements of the time for a number of periods per measurement ni.
We assume that each time measurement has the same uncertainty
.
We estimate the mean period by minimizing the
of the measurements with respect to the mean period ,
by setting the first derivative to zero:
We solve for the mean period:
To find the uncertainty on
we use the second derivative of the :
which gives us
We can estimate
from the standard deviation of the measured times with respect to the estimated mean period times the number of periods measured in measurement :These formulae give simple, well-known results for two examples:
1) All
2) All