resulting in an exponential decay
In this equation t may be time (e.g. attenuation of a circulating beam)
or length (e.g. attenuation of light in a scintillator) or any corresponding continuous variable. The
attenuation time or
attenuation length is given by 
, the time (length) over which the intensity is reduced by a factor e.
Frequently I is a discrete variable (number of particles),
and the factor 
is due to the exponential distribution of individual lifetimes. is then the expectation value of the distribution, i.e. the mean lifetime.
If the intensity at time zero is I0 and is the lifetime or attenuation time, then the average intensity over a time 
is given by 
.