Random Processes

Abstract

Let us return to the process of radioactive decay discussed above, where radium Ra disintegrates into radon Rn, by emitting α-particles. Let ξ(t) be the total number of α-particles emitted up to time t. Of course, for any 0 ≤ s ≤ t, the difference ξ(t) − ξ(s) is the number of α-particles emitted during the time interval (s, t]. As we already know, the random variable ξ(t) − ξ(s) is distributed according to the Poisson law

$$\matrix{ {P\{ \xi (t) - \xi (s) = k\} = {{{{[a(t - s)]}^k}} \over {k!}}{{\rm{e}}^{ - a(t - s)}},} & {k = 0,1, \ldots ,} \cr } $$

(1.1)

with the mean value

$$ a(t - s) = E[\xi (t) - \xi (s)] $$

which depends on the difference t − s only. We have

$$\matrix{ {a(t) = a(s) - a(t - s),} & {0 \le s \le t} \cr } < \infty ,$$

since

$$ \xi (t) = \xi (s) + [\xi (t) - \xi (s)], $$

which implies that a(t) is linear:

$$\matrix{ {a(t) = at,} & {t \ge 0.} \cr } $$

(1.2)