I am new to the concept of Poisson Process and I seem to be missing something.
Earthquakes occur in a given region in accordance with a Poisson process with rate 5 per year
a) What is the probability there will be at least two earthquakes in the first half of 2010?
b) Assuming that the event in part(a) occurs , what is the probability that there will be no earthquakes in the first 9 months of 2011?
c) Assuming that a) occurs what is the probability that there will be at least four earthquakes over the first 9 months of 2010?
Through readings I have understood that the inter-arrival times are exponentially distributed, while the distribution of events follow a Poisson Distribution.
For part a), the rate of earthquakes happening in that given region would reduce to (5/2) as we are considering the first half of the year 2010.
As the distribution of earthquakes in this given time frame is Poisson , the probability that the number of earthquakes is greater than or equal to two , would be $1-P_0-P_1$ Where $$P_i = 2.5^i \exp(-2.5) \frac{1}{i!}$$
For part(b) I think that the rate would be (5/12)*9=3.75 The probability would just be exp(-3.75)
Part (c) is the question I am getting stuck at . Assuming that at least two earthquakes happen in the first 6 months of 2010, how to we find the probability that at least four earthquakes would happen in the first 9 months? Do we have to exploit the memory less property of exponential distribution here?
Best Answer
You just have to sum over all probabilities. There are 3 possibilities you need to be concerned about:
(a) exactly 2 earthquakes in first 6 months, given that at least 2 happened in those 6 months, AND at least 2 earthquakes in next 3 months.
(b) exactly 3 earthquakes in first 6 months, given that at least 2 happened in those 6 months, AND at least 1 earthquake in next 3 months.
(c) at least 4 earthquakes in first 6 months, given that at least 2 happened in those 6 months.
The memoryless property means the AND is straight multiplication.