Poisson Processes: Properties, Inter-Arrival Times and Compound Poisson

Understand Poisson Processes

Key rules to remember

Poisson count

P(N(t) = n) = e^(−λt) (λt)ⁿ ÷ n!

Mean and variance of N(t) both equal λt.

Inter-arrival time

T ~ Exponential(λ); f(t) = λe^(−λt); P(T > t) = e^(−λt); E[T] = 1/λ; Var[T] = 1/λ²

Gaps are independent. Time to the nth event is Gamma(n, λ) with mean n/λ.

Memoryless property

P(T > s + t | T > s) = P(T > t)

Holds for the exponential only, among continuous distributions.

Superposition

N₁(t) + N₂(t) is Poisson process with rate λ₁ + λ₂

Requires independent processes.

Thinning

Type 1 events: rate pλ; type 2 events: rate (1 − p)λ

Each event classified independently. The two resulting processes are independent.

Compound Poisson mean

E[S(t)] = λt E[X]

X is a single claim size.

Compound Poisson variance

Var[S(t)] = λt E[X²]

Uses the second moment, not the variance, of X.

Compound Poisson MGF

M_S(r) = exp{λt (M_X(r) − 1)}

Useful for skewness and for proving closure results.

Third central moment

E[(S − E[S])³] = λt E[X³]

Gives the skewness as λt E[X³] ÷ (λt E[X²])^(3/2).

How to solve Poisson Processes questions

1. 1Write down the rate λ and the time unit. Convert all times to the same unit before doing anything else.
2. 2Decide what is asked: a count in an interval (Poisson), a waiting time (exponential or gamma), a split by type (thinning), or a total claim amount (compound).
3. 3For counts, compute the mean λt for the interval given and use the Poisson probability formula. Use independence for disjoint intervals.
4. 4For waiting times, use P(T > t) = e^(−λt). For the nth event, sum n gaps or use the gamma, or convert to a count: the nth event occurs after t if N(t) ≤ n − 1.
5. 5For several event types, use thinning to get each rate, or superposition to add rates. Check the independence conditions are met.
6. 6For aggregate claims, find E[X] and E[X²] from the claim distribution, then apply the compound formulas for mean and variance.
7. 7If a conditional probability is asked, use independent increments to separate intervals, or use the memoryless property to restart the clock.
8. 8State the result with units and check it is sensible: probabilities between 0 and 1, variance at least zero.

Quickest way: Rate-scaling shortcut

Common mistakes in Poisson Processes

• Using Var[S] = λt Var[X] for a compound Poisson process.

  Students mix this up with the general compound formula E[N]Var[X] + Var[N]E[X]².

  Fix: For Poisson N, Var[N] = E[N] = λt, so the result simplifies to λt E[X²]. Always use the second moment.

• Forgetting to change the time unit, for example using a monthly rate for a yearly interval.

  The rate and the interval are given in different parts of the question.

  Fix: Write the unit next to λ and convert the interval length before computing λt.

• Treating the mean wait as λ instead of 1/λ.

  The exponential density is λe^(−λt), so λ is easy to confuse with the mean.

  Fix: Remember λ is a rate (events per unit time), so the mean gap is 1/λ.

• Applying independence to overlapping intervals, such as N(2) and N(5).

  Students assume all counts in a Poisson process are independent.

  Fix: Only disjoint intervals are independent. Split N(5) = N(2) + (N(5) − N(2)) and use the independent pieces.

• Saying the nth arrival time is exponential.

  The single gap is exponential, and students carry this over.

  Fix: The nth arrival time is a sum of n independent exponentials, so it is Gamma(n, λ).

• Adding rates for thinned processes instead of multiplying by p.

  Superposition and thinning are learned together and get confused.

  Fix: Merging independent processes adds rates. Splitting one process by probability p multiplies its rate by p.

Worked examples