Question

Earthquakes occur in the United States according to a Poisson process having a rate of 0.25 per day. Suppose we begin counting earthquakes at some point in time. a. What is the probability that 6 earthquakes occur in July 2050? b. On average, when will the 50th earthquake occur? 25 Example 4.5 What is the probability that 2 or more earthquakes occur over a 50-day period? c. d. What is the probability that it takes more than 10 days until the 3rd earthquake? e. On average, how many earthquakes will occur in 2052? f. On average, how long will it be between the 100th and 101 earthquakes? Example 4.5 g. What is the probability of no earthquakes over a 5-day period? h. What is the probability of no earthquakes over 5 separate 1-day periods? i. What is the probability of no earthquakes in the period June 2050 August 2050? j. What is the probability of 24 earthquakes in the period June 2050 August 2050?

Answer 1

a) July, there are 31 days, therefore avg number of earthquakes in 31 days is computed as:

= 31*0.25 = 7.75

The probability here is computed as:

7.756 7.75 = 0.1296

Therefore 0.1296 is the required probability here.

b) On average the 50th earthquake would occur on: 50/0.25 = 200^th day

c) The average number of earthquakes in a 50 day period is given as: 0.25*50 = 12.5

Therefore the probability here is computed as:

12.5 -12.5єー12.5 = 0.999950 PlX > 2) = 1-PlX = 0)-P(X = 1) = 1-e-

d) Probability that it takes more than 10 days for the third earthquake to occur is computed as:

= Probability of less than 3 earthquakes in 10 days

P(X < 3) = 1-P(X = 0) _ P(X = 1) = 1-e-2.5-2.5e-25ー0.7127

Therefore 0.7127 is the required probability here.