In a Poisson Process, counts in disjoint time intervals are independent.

In a Poisson Process, counts in disjoint time intervals are independent.

• A. Counts in disjoint intervals are independent.
• B. Counts in disjoint intervals are dependent.
• C. The process has three outcomes per trial.
• D. It models the time between events rather than counts.

Answer: (A)

Independent increments are at the heart of a Poisson process: the number of events that happen in non-overlapping time intervals are independent random variables. Because events occur at a constant average rate and interarrival times are memoryless, the counts in each interval follow a Poisson distribution with a mean proportional to the interval length, and knowing the count in one interval tells you nothing about the count in a different disjoint interval. That’s why the statement is true. The alternative ideas either misstate the model (the process isn’t about three outcomes per trial) or contradict the defining property (counts in disjoint intervals are not dependent), or mix in the time-between-events perspective without addressing the independence of counts in disjoint intervals.

Independent increments are at the heart of a Poisson process: the number of events that happen in non-overlapping time intervals are independent random variables. Because events occur at a constant average rate and interarrival times are memoryless, the counts in each interval follow a Poisson distribution with a mean proportional to the interval length, and knowing the count in one interval tells you nothing about the count in a different disjoint interval. That’s why the statement is true. The alternative ideas either misstate the model (the process isn’t about three outcomes per trial) or contradict the defining property (counts in disjoint intervals are not dependent), or mix in the time-between-events perspective without addressing the independence of counts in disjoint intervals.