Editing Mean time between failures (section)

=== MTBF considering censoring === 

In fact the MTBF counting only failures with at least some systems still operating that have not yet failed underestimates the MTBF by failing to include in the computations the partial lifetimes of the systems that have not yet failed.  With such lifetimes, all we know is that the time to failure exceeds the time they've been running.  This is called [[Censoring (statistics)|censoring]].  In fact with a parametric model of the lifetime, the [[Censoring (statistics)#likelihood|likelihood for the experience on any given day is as follows]]:

:<math>L = \prod_i \lambda(u_i)^{\delta_i} S(u_i)</math>, 

where 
:<math>u_i</math> is the failure time for failures and the censoring time for units that have not yet failed, 
:<math>\delta_i</math> = 1 for failures and 0 for censoring times, 
:<math>S(u_i)</math> = the probability that the lifetime exceeds <math>u_i</math>, called the survival function, and 
:<math>\lambda(u_i) = f(u)/S(u)</math> is called the [[Failure rate#hazard function|hazard function]], the instantaneous force of mortality (where <math>f(u)</math> = the probability density function of the distribution).  

For a constant [[exponential distribution]], the hazard, <math>\lambda</math>, is constant.  In this case, the MBTF is 

:MTBF = <math>1 / \hat\lambda = \sum u_i / k</math>, 

where <math>\hat\lambda</math> is the maximum likelihood estimate of <math>\lambda</math>, maximizing the likelihood given above and <math>k = \sum \sigma_i</math> is the number of uncensored observations.  

We see that the difference between the MTBF considering only failures and the MTBF including censored observations is that the censoring times add to the numerator but not the denominator in computing the MTBF.<ref>{{cite Q|Q98961801}}<!-- Likelihood Construction, Inference for Parametric Survival Distributions -->.</ref>