Simple Nature - Light and Matter

Average lifetime example 6
You might think that the half-life would also be the average life-
time of an atom, since half the atoms’ lives are shorter and half
longer. But the half whose lives are longer include some that sur-
vive for many half-lives, and these rare long-lived atoms skew the
average. We can calculate the average lifetime as follows:

(average lifetime) =

∫∞

0

t D(t) dt

Using the convenient base-eform again, we have

(average lifetime) =

ln 2

t 1 / 2

∫∞

0

texp

(

−

ln 2

t 1 / 2

t

)

dt.

This integral is of a form that can either be attacked with in-
tegration by parts or by looking it up in a table. The result is∫
xecxdx = xcecx−c^12 ecx, and the first term can be ignored for
our purposes because it equals zero at both limits of integration.
We end up with

(average lifetime) =

ln 2

t 1 / 2

(

t 1 / 2

ln 2

) 2

=

t 1 / 2

ln 2

= 1.443t 1 / 2 ,

which is, as expected, longer than one half-life.

Section 13.1 Rules of randomness 869