MAST30001 Chap.3 Poisson Processes
Poisson Processes
Poisson processes describe random event streams with independent increments and a constant rate. The same process can be viewed through counts N(t), arrival epochs T(k), or exponential interarrival times. This chapter develops those translations and then studies operations that preserve the model: superposition, independent thinning and compound Poisson sums.
Unit discipline is central because rates, elapsed times and expected counts play different roles. The choice of representation should follow the event: fixed-window questions favour counts, numbered-arrival questions favour epochs, and gaps favour exponential waiting times. Structural operations are valid only when their independence and constant-probability assumptions are stated.
What this chapter covers
- 01
Definition through independent increments and Poisson counts
- 02
Count, arrival-time and interarrival descriptions
- 03
Exponential waiting times and gamma arrival epochs
- 04
Superposition of independent streams
- 05
Independent thinning and marked arrivals
- 06
Compound Poisson sums and PASTA
Worked example · free
Combine a count, wait and thinning calculation
- 1The 30-minute count mean is 6(0.5)=3.
- 2The exact-three probability is exp(-3)3^3/3! = 0.2240.
- 2Twenty minutes is one third hour, so the survival probability is exp(-2)=0.1353.
- 1Independent thinning gives urgent rate 6(0.25)=1.5 per hour.
Key terms
- Independent Increments
- Counts accumulated over disjoint time intervals are independent random variables.
- Arrival Epoch
- The time at which a specified numbered event occurs in a counting process.
- Interarrival Time
- The elapsed time between two successive events in an arrival process.
- Superposition
- The combined stream formed by adding independent Poisson processes, with rate equal to the sum of rates.
- Poisson Thinning
- Independent random labelling of arrivals that creates independent Poisson branches with multiplied rates.
- Compound Poisson Process
- A random sum whose number of independent marks by time t follows a Poisson process.
- Stationary Increments
- The distribution of an increment depends on interval length rather than its position on the time axis.
- Marked Arrival
- An event paired with a random label or magnitude, such as a type, cost or claim size.
Poisson Processes FAQ
When should I use a count rather than a waiting time?
Use a count when the event asks how many arrivals occur by a fixed time. Use an arrival time when it asks when a numbered event occurs. The identity T(k) at most t exactly matches N(t) at least k, so choose whichever distribution is easier.
Why does thinning preserve the Poisson model?
Each arrival receives an independent label with a constant probability, so the retained and removed counts inherit independent increments and Poisson distributions with rates p lambda and one minus p times lambda. Deterministic or history-dependent splitting does not satisfy the same conclusion.
How do I calculate a compound Poisson variance?
Condition on the Poisson count and account for both sources of variability: the random number of marks and the marks themselves. For independent marks X and count mean m, the aggregate variance is m times the second moment E[X squared], not merely m times Var(X).
How do I convert an hourly rate for a short interval?
Express rate and time in compatible units before multiplying. A rate of twelve per hour is 0.2 per minute, so a ten-minute count has mean two. The resulting rate-times-time product is dimensionless and represents an expected count, which is the parameter used in the Poisson probability mass function.
When can independent Poisson streams be combined?
Superposition applies when the component processes are independent Poisson processes on the same time scale. The combined count is Poisson with the sum of component rates. If streams share events or respond to a common random environment, independence may fail and adding their nominal rates alone does not justify the conclusion.
What is the quickest waiting-time plausibility check?
For an exponential waiting time, the survival probability must decrease as the threshold grows and equal one at threshold zero. Its mean is the reciprocal of the rate, so a higher event rate should produce a shorter typical wait. These direction checks often expose a missing minus sign or unit conversion.
Exam move
Practise translating among counts, arrival epochs and interarrival gaps before memorising formulas. Put units beside every rate, convert the elapsed time, and check that lambda times t is an expected count. Build a small diagram for superposition and thinning, labelling every branch rate and verifying that it recombines correctly. For compound sums, condition on the count and separate count variability from mark variability.
Use limiting checks: a zero-length interval should produce no arrivals with probability one, and longer waiting thresholds should have smaller survival probabilities. Create a translation table with one scenario written as a count event, an arrival-epoch event and an interarrival sum. Then practise deciding which side gives the shorter calculation.
For each thinning question, label whether selection is independent, whether its probability is constant and whether discarded and retained streams are both required. For every compound process, write the units of one mark and multiply them by the expected count only after confirming count-mark independence.
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