Poisson Process



Consider a certain event that occurs consecutively at random time points
$$0 < {T}_{1} < {T}_{2} < \cdots < {T}_{n} < \cdots \,.$$
For convenience, we write T 0=0. Denote
$${X}_{i} = {T}_{i} - {T}_{i-1}$$
for i=1,2,⋯as the interarrival times. Let N(t) be the total number of occurrences of the event in the time interval (0,t], t>0, i.e.,
$$N(t) =\sup \{ n :\ {T}_{n} \leq t\},\ \ \mathrm{for}\ \ t > 0.$$
We call N(t),t≥0 a counting process. For each t>0, N(t) could represent the number of accidents at a particular intersection, the number of times a computer breaks down, or similar counts.


Poisson Process Renewal Process Soccer Player Moment Generate Function Interarrival Time 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBowling Green State UniversityBowling GreenUSA
  2. 2.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  3. 3.Department of MathematicsCalifornia State University StanislausTurlockUSA

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