1. Priliminary Definitions
Def : A stochastic process is a collection of random variable (RV) indexed by time .
- If T is continuous set, the process is a continuous time stochastic process (e.g., Poisson Process)
- If T is countable, then the process is a discrete time stochastic process (e.g., Markov chain)
- (that is, N(t) is non-negative integer)
- If , the ( that is, N(t) is non-decreasing in t)
- For , is the number of events occurring in the time interval .
2. Definition of Poisson Process
Definition 1: A Poisson process is a counting process with rate , if:
- N(0) = 0;
- The Process has independent increments
- The number of events in any interval of length t is a Poisson RV with mean .
Definition 2: A Poisson process is a counting process with rate if
- N(0) = 0
- The process has stationary increments
- The process has independent increments
- (# of events approximately proportional to the length of interval)
- (can’t have 2 or more events at the same time — “orderliness”)
- Eliminate assumption 2, Non-stationary Poisson Process
- Eliminate assumption 3, Mixture of Poisson Process (choose randomly, then run a Poisson process)
- Eliminate assumption 5, compound Poisson Process
3. Conditional Distribution of Event Times