Given an assembly of elements, the number of which decreases ultimately to zero, the lifetime (also called the mean lifetime) is a certain number that characterizes the rate of reduction ("decay") of 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. In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). ...

Typically, the notion of mean lifetime is used in connection with exponential decay. The remainder of this article confines itself to this particular decay pattern. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

Mean lifetime in exponential decay

As you will see derived below, the mean lifetime τ of elements in an exponentially decaying assembly is equal to the reciprocal of the decay constant (cf. Exponential decay). Thus, it is the time needed for the assembly to be reduced by a factor of e. It is related to the half-life t_{1 / 2} thus: Tau (upper case Τ, lower case Ï„) is the 19th letter of the Greek alphabet. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... The title given to this article is incorrect due to technical limitations. ... Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ...

Derivation

For purposes of this derivation, the following terminology and notation (some of it introduced above) will be useful:

• A large (but decreasing) number of elements is grouped into an assembly.
• The individual lifetime of an element is the time elapsed between some reference time and the removal of that element from the assembly.
• The total lifetime (T) is the sum of the individual lifetimes.
• The mean lifetime (τ) is the arithmetic mean of the individual lifetimes.
• The population (N) is the number of elements in the assembly at any given time.
• The initial population (N₀) is the population at some initial reference time.

In exponential decay, the population is governed by the following formula:

During a differential period of time dt, the population shrinks slightly. The (positive) number of elements leaving the assembly is equal to the negative of the change in population: In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ...

If dt is taken at some time t, the individual lifetimes of the elements are simply equal to t. These elements' contribution to the total lifetime is thus given by:

Integration by parts yields: In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...