Events occur according to a Poisson process with rate λ = 3 per hour. Given that exactly 1 event occurred in [0,2], the conditional distribution of the event's time is:
AExponential with rate 3
BUniform on [0,2]
CNormal with mean 1 and variance 1/3
DThe event occurs at time 1 with probability 1 (deterministic)
This is a fundamental property of the Poisson process: given N(t) = n events in [0,t], the event times are distributed as the order statistics of n independent Uniform([0,t]) random variables. With n = 1, the single event time is Uniform on [0,2]. This follows from the homogeneity of the Poisson process — events are equally likely to occur at any moment within the interval. The result generalizes: given N(t) = n, the n arrival times have the same joint distribution as n i.i.d. Uniform([0,t]) random variables, sorted.
Question 2 True / False
The sum of two independent Poisson processes with rates λ₁ and λ₂ is again a Poisson process, with rate λ₁ + λ₂.
TTrue
FFalse
Answer: True
If N₁ and N₂ are independent Poisson processes with rates λ₁ and λ₂, then N(t) = N₁(t) + N₂(t) has independent stationary increments (inherited from the independence of N₁ and N₂), and N(t) - N(s) = (N₁(t)-N₁(s)) + (N₂(t)-N₂(s)) is the sum of two independent Poisson random variables, which is Poisson(λ₁(t-s) + λ₂(t-s)) = Poisson((λ₁+λ₂)(t-s)). This superposition property makes Poisson processes the natural model for merged independent event streams.
Question 3 Short Answer
Explain why the exponential distribution of inter-arrival times is equivalent to the memoryless property of the Poisson process.
Think about your answer, then reveal below.
Model answer: The memoryless property states P(T > t+s | T > t) = P(T > s) — knowing that no event has occurred for t time units doesn't change the distribution of the remaining waiting time. The exponential distribution is the unique continuous distribution with this property: P(T > t+s | T > t) = e^{-λ(t+s)}/e^{-λt} = e^{-λs} = P(T > s). This means the process 'restarts' after each moment — the future is independent of the past given that no event has occurred. The memoryless property characterizes the Poisson process among all counting processes with stationary increments.
The memoryless property is both the defining intuition and the characterizing theorem for exponential inter-arrivals. The geometric distribution plays the same role in discrete time: it is the unique memoryless discrete distribution, and the Poisson process is the continuous-time limit of Bernoulli trials with geometric inter-arrivals.
Question 4 True / False
A compound Poisson process X(t) = Σᵢ₌₁^{N(t)} Yᵢ, where N is Poisson(λt) and Yᵢ are i.i.d., has E[X(t)] = λt·E[Y₁] and Var(X(t)) = λt·E[Y₁²].
TTrue
FFalse
Answer: True
By the law of total expectation: E[X(t)] = E[N(t)]·E[Y₁] = λt·E[Y₁]. By the law of total variance: Var(X(t)) = E[N(t)]·Var(Y₁) + Var(N(t))·(E[Y₁])² = λt·Var(Y₁) + λt·(E[Y₁])² = λt·E[Y₁²]. The compound Poisson process generalizes the Poisson process by allowing each event to carry a random magnitude Yᵢ — it models aggregate insurance claims, total demand in a queue, or cumulative jump sizes in a financial model.