# Random Processes

• Yu. A. Rozanov
Part of the Mathematics and Its Applications book series (MAIA, volume 344)

## 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 ≤ st, 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 ts 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)

## Keywords

Brownian Motion Random Process Poisson Process Conditional Probability Density Stationary Probability Distribution
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