Suppose X has a Poisson distribution with parameter a. Which of the following st

Question

Suppose X has a Poisson distribution with parameter a. Which of the following statements are true? The probability of 1 event in an interval of length 10 units of time is the same as the probability of 2 events in an interval of length 20 units of time. O The probability of 2 events in an interval of length 10 units of time is the same as the probability of 2 total events in two separate intervals of length 5 units of time. The probability of 2 events in an interval of length 10 units of time is the same as the probability of 1 event in one interval of length 5 minutes and 1 event in a second interval of length 5 minutes. Onone of the above

Explanation / Answer

Answer: Option b: The probability of 2 events in an interval of 10 units of time is the same as the probability of 2 total events in two separate intervals of 5 units of time each.

Explanation: The Poisson distribution is a memoryless distribution. This means that it does not matter when we start the clock to consider one interval of time. We can regard each interval of time as a separate event. Thus, in this option, there are 10 events. All the ten events are independent of each other. It does not matter whether the events are consecutive in time or in groups or completely separate, provided that there is no overlap between any pair of intervals. Thus, the probability that it happens exactly twice across all the ten events is also the same.

We can even write out the probabilities as:
P(exactly two events in one of the intervals + exactly zero events in remaining nine intervals) +
P(exactly one event in two of the intervals + exactly zero events in remaining eight intervals)

From this, it becomes clear that the probabilities will not change when the 10 intervals are continuous or separated in groups of 5 or any other combination.

We can also see why options a) and c) are incorrect when we write out their expansion.

Option a)

P(1 event across 10 intervals) = P(1 event in 1 interval and 0 events in 9 intervals)

P(2 events across 20 intervals) = P(2 events in 1 interval and 0 events in 19 intervals) + P(1 event in 2 intervals and 0 events in 18 intervals)

Clearly, probabilities are not equal.

Option c)

P(2 events across 10 intervals) = P(2 events in 1 interval and 0 events in 9 intervals) + P(1 event in 2 intervals and 0 events in 8 intervals)

P(1 event across 5 intervals)2 = P(1 event in 1 interval and 0 events in 4 intervals)2

Again, it is clear that probabilities are not equal.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.