Editing Dead time (section)

== Analysis ==
It will be assumed that the events are occurring randomly with an average frequency of ''f''. That is, they constitute a [[Poisson process]]. The probability that an event will occur in an infinitesimal time interval ''dt'' is then ''f dt''. It follows that the probability ''P(t)'' that an event will occur at time ''t''&nbsp; to ''t+dt'' with no events occurring between ''t=0'' and time ''t''&nbsp; is given by the [[exponential distribution]] (Lucke 1974, Meeks 2008):

:<math>P(t)dt=fe^{-ft}dt\,</math>

The expected time between events is then

:<math>\langle t \rangle = \int_0^\infty tP(t)dt = 1/f</math>

=== Non-paralyzable analysis ===
For the non-paralyzable case, with a dead time of <math>\tau</math>, the probability of measuring an event between <math>t=0</math> and <math>t=\tau</math> is zero. Otherwise the probabilities of measurement are the same as the event probabilities. The probability of measuring an event at time ''t'' with no intervening measurements is then given by an exponential distribution shifted by <math>\tau</math>:

:<math>P_m(t)dt=0\,</math> for <math>t\le\tau\,</math>

:<math>P_m(t)dt=\frac{fe^{-ft}dt}{\int_\tau^\infty fe^{-ft}dt} = fe^{-f(t-\tau)}dt</math> for <math>t>\tau\,</math>

The expected time between measurements is then

:<math>\langle t_m \rangle = \int_\tau^\infty tP_m(t)dt = \langle t \rangle+\tau</math>

In other words, if <math>N_m</math> counts are recorded during a particular time interval <math>T</math> and the dead time is known, the actual number of events (''N'') may be estimated by <ref>{{cite thesis |last1=Patil |first1=Amol |title=Dead time and count loss determination for radiation detection systems in high count rate applications |type=PhD Thesis |date=2010 |page=2148 |url=https://scholarsmine.mst.edu/cgi/viewcontent.cgi?article=3150&context=doctoral_dissertations}}</ref>

:<math>N \approx \frac{N_m}{1 - N_m \frac{\tau}{T}}</math>

If the dead time is not known, a statistical analysis can yield the correct count. For example, (Meeks 2008), if <math>t_i</math> are a set of intervals between measurements, then the <math>t_i</math> will have a shifted exponential distribution, but if a fixed value ''D'' is subtracted from each interval, with negative values discarded, the distribution will be exponential as long as ''D'' is greater than the dead time <math>\tau</math>. For an exponential distribution, the following relationship holds:

:<math>\frac{\langle t^n \rangle}{\langle t \rangle^n} = n!</math>

where ''n'' is any integer. If the above function is estimated for many measured intervals with various values of ''D'' subtracted (and for various values of ''n'') it should be found that for values of ''D'' above a certain threshold, the above equation will be nearly true, and the count rate derived from these modified intervals will be equal to the true count rate.