A counting process is said to be a non-homogeneous Poisson process with intensity function if: (ii) has independent increments Prove that follows a Poisson distribution with parameter .

Probability Theory Solution – Problem 16: Prove that follows a Poisson distribution with parameter

Question

A counting process $\{N(t),t\geqslant0\}$ is said to be a non-homogeneous Poisson process with intensity function $\lambda\left(t\right),t\geqslant0$ if:

$(i)\ N(0)=0$

(ii) $N(t),t\geqslant0$ has independent increments

$(i i i)\ P\{N(t+h)-N(t)=1\}=\lambda\left(t\right)h+o\left(h\right)$

$(i v)\ P\{N(t+h)-N(t)\geqslant2\}=o\left(h\right).$

Prove that $N(t+s)-N(t)$ follows a Poisson distribution with parameter $\int_{t}^{t+s}\lambda\left(u\right)\,d u$ .

Step-by-step solution

Step 1. Partition the interval $[t,t+s]$ into $n$ equal subintervals and set $t_k=t+k\,s/n$ . By the independent increments property, $N(t+s)-N(t)=\sum_{k=0}^{n-1}\big(N(t_{k+1})-N(t_k)\big).$ Step 2. By definition, for each subinterval, $P\{N(t_{k+1})-N(t_k)=1\}=\lambda(t_k)\frac{s}{n}+o\!\left(\frac{s}{n}\right), \qquad P\{N(t_{k+1})-N(t_k)\ge2\}=o\!\left(\frac{s}{n}\right),$ so each subinterval produces at most one jump with high probability. Step 3. Summing these probabilities yields $\sum_{k=0}^{n-1}\left(\lambda(t_k)\frac{s}{n}+o\!\left(\frac{s}{n}\right)\right) \to \int_t^{t+s}\lambda(u)\,du,$ which is a Riemann sum converging to the definite integral. Step 4. Since the subintervals are independent and each admits at most one jump with small probability, $N(t+s)-N(t)$ is a sum of independent Bernoulli random variables. Letting $n\to\infty$ , by the Poisson limit theorem, $P\{N(t+s)-N(t)=k\} =\exp\!\left(-\int_t^{t+s}\lambda(u)\,du\right) \frac{\big(\int_t^{t+s}\lambda(u)\,du\big)^k}{k!}.$

Final answer

QED.

Marking scheme

Here is the grading rubric for the problem.

1. Checkpoints (max 7 pts total)

Score exactly one chain | take the maximum subtotal among chains; do not add points across chains.

Chain A: Partition and Limit Method [Official Approach]

[1 pt] Partitioning: Partition the time interval $[t, t+s]$ into $n$ equal (or arbitrary) subintervals and explicitly express the total increment $N(t+s)-N(t)$ as the sum of the increments over each subinterval.
[1 pt] Local Probability Analysis: Correctly invoke conditions (iii) and (iv) to show that in the $k$ -th subinterval, the probability of exactly one arrival is approximately $\lambda(t_k)\Delta t$ (or $\lambda(t_k)\frac{s}{n}$ ), and the probability of two or more arrivals is $o(\Delta t)$ .
[2 pts] Riemann Sum Limit: Sum the jump probabilities over all subintervals and correctly identify the limit as the definite integral $\int_{t}^{t+s}\lambda(u)\,du$ .
*Note: If only the summation is written without taking the limit to obtain the integral, award 1 pt.*
[3 pts] Distribution Conclusion: Combine the independent increments property (ii) with the Poisson Limit Theorem (or the limiting distribution of a sequence of Bernoulli trials) to conclude that the sum follows a Poisson distribution with the above integral as its parameter.
*Note: If only the expectation is shown to equal the integral without justifying why the distribution is Poisson, no credit is awarded for this item.*

Chain B: ODE / Probability Generating Function Method

[2 pts] Establishing the Difference Relation: Define $P_n(\tau) = P(N(t+\tau)-N(t)=n)$ or introduce the characteristic function/generating function, and use the total probability formula together with conditions (iii)-(iv) to establish a relation between times $\tau$ and $\tau+h$ (e.g., $P_n(\tau+h) \approx P_n(\tau)(1-\lambda h) + P_{n-1}(\tau)\lambda h$ ).
[2 pts] Derivation of the ODE: Let $h \to 0$ and correctly derive the Kolmogorov forward equations $P_n'(\tau) = -\lambda(t+\tau)P_n(\tau) + \lambda(t+\tau)P_{n-1}(\tau)$ (or the corresponding PDE for the generating function).
*Key point: The intensity function must be time-dependent $\lambda(t+\tau)$ , not a constant.*
[1 pt] Solving the Zeroth-Order Term: Correctly solve $P_0(\tau) = \exp\left(-\int_t^{t+\tau}\lambda(u)\,du\right)$ or the logarithmic part of the characteristic function.
[2 pts] Induction or Inversion: Solve for the general term $P_n(\tau)$ by mathematical induction, or invert/Taylor-expand the characteristic function to obtain the complete Poisson distribution.

Total (max 7)

2. Zero-credit items

Transcription Only: Merely copying the definitions (i)-(iv) from the problem statement without any substantive derivation.
Circular Reasoning: Directly assuming that $N(t)$ is a non-homogeneous Poisson process and invoking its known properties as the conclusion, rather than proving the result from the microscopic conditions (i)-(iv).
Unsupported Assertion: Writing down the final formula $P(N(t+s)-N(t)=k) = \dots$ without any intermediate partition-and-sum or differential equation argument.

3. Deductions

[Cap at 3/7] Constant Intensity Fallacy: Assuming $\lambda(t) \equiv \lambda$ (constant) throughout the proof. This drastically simplifies the core difficulty of the problem (handling the variable-limit integral), so even if the logic is otherwise sound, the total score is capped at 3.
[-1] Missing $o(h)$ Terms: Completely ignoring the $o(h)$ terms or handling them improperly (e.g., writing $=0$ instead of $\approx$ or $o(h)$ ) when deriving the differential equation or taking limits, thereby compromising mathematical rigor.
[-1] Incorrect Integration Limits: Writing the wrong limits of integration in the final result or derivation (e.g., $\int_0^s \lambda(u)du$ without accounting for the shift by $t$ ; the correct form is $\int_t^{t+s}\lambda(u)du$ or equivalently $\int_0^s \lambda(t+u)du$ ).
[-1] Logic Gap: In Chain A, after computing the probability sum, directly asserting the Poisson distribution without mentioning that the subinterval increments are independent or invoking the rare-event principle.

Probability Theory – Problem 16: Prove that follows a Poisson distribution with parameter