A Poisson process is a mathematical model for random events that occur over time, such as phone calls arriving, radioactive decays, or customers entering a store. It is useful when events happen one at a time, independently, and at a steady average rate. Instead of predicting the exact time of the next event, the model predicts probabilities for counts and waiting times.
This makes it a powerful tool in physics, engineering, biology, and service systems.
The key parameter is the rate λ, which means the average number of events per unit time. In any time interval of length t, the number of events follows a Poisson distribution with mean λt. The gaps between neighboring events follow an exponential distribution, which gives the probability of waiting a certain amount of time until the next event.
The exponential waiting time is memoryless, meaning that if no event has happened yet, the future waiting time has the same distribution as it did at the start.
Key Facts
- Rate parameter: λ = average number of events per unit time.
- Event count in time t: N(t) ~ Poisson(λt).
- Poisson probability: P(N(t) = k) = e^(-λt)(λt)^k / k!.
- Mean and variance of counts: E[N(t)] = λt and Var(N(t)) = λt.
- Waiting time to the next event: T ~ Exponential(λ), with f(t) = λe^(-λt) for t ≥ 0.
- Memoryless property: P(T > s + t | T > s) = P(T > t) = e^(-λt).
Vocabulary
- Poisson process
- A random process that counts independent events occurring over time at a constant average rate.
- Rate parameter
- The value λ that gives the average number of events expected per unit time.
- Poisson distribution
- A probability distribution for the number of events in a fixed interval when events follow a Poisson process.
- Exponential distribution
- A probability distribution that describes the waiting time until the next event in a Poisson process.
- Memoryless property
- The property that the probability of waiting longer does not depend on how long you have already waited.
Common Mistakes to Avoid
- Using λ as the expected count for every interval is wrong because the expected count over time t is λt, not just λ.
- Assuming events must be evenly spaced is wrong because a Poisson process has random gaps, often with clusters and long quiet periods.
- Using a Poisson distribution for waiting times is wrong because counts are Poisson but waiting times are exponential.
- Ignoring the independence assumption is wrong because the Poisson process model requires that events in separate time intervals do not influence each other.
Practice Questions
- 1 A sensor detects particles at an average rate of λ = 4 particles per minute. What is the probability that exactly 6 particles are detected in 2 minutes?
- 2 Calls arrive at a help desk according to a Poisson process with rate λ = 3 calls per hour. What is the probability that the waiting time until the next call is more than 20 minutes?
- 3 A bus stop is sometimes modeled with a Poisson process for bus arrivals. Explain why the memoryless property may fail if buses are actually scheduled every 10 minutes.