Random Processes

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


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
$$\matrix{ {P\{ \xi (t) - \xi (s) = k\} = {{{{[a(t - s)]}^k}} \over {k!}}{{\rm{e}}^{ - a(t - s)}},} & {k = 0,1, \ldots ,} \cr } $$
with the mean value
$$ a(t - s) = E[\xi (t) - \xi (s)] $$
which depends on the difference ts only. We have
$$\matrix{ {a(t) = a(s) - a(t - s),} & {0 \le s \le t} \cr } < \infty ,$$
$$ \xi (t) = \xi (s) + [\xi (t) - \xi (s)], $$
which implies that a(t) is linear:
$$\matrix{ {a(t) = at,} & {t \ge 0.} \cr } $$


Brownian Motion Random Process Poisson Process Conditional Probability Density Stationary Probability Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Yu. A. Rozanov
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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