Editing Exponential decay (section)

=== Derivation of the mean lifetime ===
Given an assembly of elements, the number of which decreases ultimately to zero, the '''mean lifetime''', <math>\tau</math>, (also called simply the '''lifetime''') is the [[expected value]] of the amount of time before an object is removed from the assembly.  Specifically, if the ''individual lifetime'' of an element of the assembly is the time elapsed between some reference time and the removal of that element from the assembly, the mean lifetime is the [[arithmetic mean]] of the individual lifetimes.

Starting from the population formula

:<math>N = N_0 e^{-\lambda t}, \,</math>

first let ''c'' be the normalizing factor to convert to a [[probability density function]]:

:<math>1 = \int_0^\infty c \cdot N_0 e^{-\lambda t}\, dt = c \cdot \frac{N_0}{\lambda}</math>

or, on rearranging,

:<math>c = \frac{\lambda}{N_0}.</math>

Exponential decay is a [[scalar multiplication|scalar multiple]] of the [[exponential distribution]] (i.e. the individual lifetime of each object is exponentially distributed), which has a [[Exponential distribution#Properties|well-known expected value]].  We can compute it here using [[integration by parts]].

:<math>\tau = \langle t \rangle = \int_0^\infty t \cdot c \cdot N_0 e^{-\lambda t}\, dt = \int_0^\infty \lambda t e^{-\lambda t}\, dt = \frac{1}{\lambda}.</math>