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binomial#
View page sourceBinomial Distribution#
In some cases, the dismod_at measured value meas_value is closely approximated by a binomial distribution. In this section we derive an approximation for the corresponding measurement standard deviation meas_std .
Notation#
n |
number of samples |
p |
probability of success for each sample |
k |
number of success events |
r |
success ratio; i.e., r = k / n |
\(\sigma\) |
approximate standard deviation of the success ratio r |
Variance#
The variance of \(k\) is given by
Converting from \(k\) to \(r\) we have
When the data is binomial, we measure k and know n, but we do not know p . Using a uniform prior on p, the expected value of p given k is
See Estimation of Parameters for a binomial distribution.
Approximation#
Using this \(\B{E} [ p | k ] = (k + 1) / (n + 2)\) as an approximation for p, the corresponding approximation for the variance give the number of success events k is
If r is the dismod_at measured value meas_value , the corresponding standard deviation meas_std is the square root of the variance approximation above; i.e.,
In some cases r / T is the dismod_at measured value, where T is the total time for the experiment. In this case the corresponding standard deviation is \(\sigma / T\) .
Example#
The user_binomial.py file contains an example and test of standard deviation approximation above.