Question

upper quarter. 2. During the years 1998–2012, a total of 29 earthquakes
of magnitude greater than 6.5 have occurred in Papua New Guinea. Assume that the time
spent waiting between earthquakes is
exponential. Do in R (Practice and check
with calculator) a. What is the probability that the next earthquake
occurs within the next three months?
b. Given that six months has passed without an
earwhat is the pr thquake in Papua New Guinea, obability that the next three
months will be free of earthquakes? c. What is the probability of zero
earthquakes occurring in 2014? d. What is the probability that at least two
earthquake distributed with a mean of 100 and

Answer 1

2.

Time difference between 1998 to 2012 is 2012-1998 = 14

Average number of earthquakes per year = 29 / 14 = 2.0714 per year = 2.0714/12 per month = 0.1726 per month

Let T be the time interval between two earthquakes. Then T ~ Exp( = 2.0714 per year = 0.1726 per month)

a.

Using the R script, pexp(3, 0.1726)

Probability that the next earthquake occurs within the next three months = P(T 3 months)

= 0.4041701

b.

Given 6 months passed without an earthquake, probability that the next three months will be free from earthquakes

= P(T > 9 | T > 6)

= P(T > 3) (By Memoryless property of Exponential distribution)

= 0.5958299 - Using R script, pexp(3, 0.1726, lower.tail = FALSE)

c.

We know that T ~ Exp( = 2.0714 per year)

Probability of zero earthquakes occurring in 2014 = P(T > 1)

= 0.1260092    - Using R script, pexp(1, 2.0714, lower.tail = FALSE)

d.

We know that since the time interval between two earthquakes follow exponential distribution, number of earthquakes within specified time period will follow Poisson distribution. Let X be the number of earthquakes within a year. Then X ~ Poisson(2.0714 per year)

Probability that at least two earthquake will occur in 2014 = P(X 2) = P(X > 1)

= 0.6129752     - Using R script, ppois(1, 2.0714, lower.tail = FALSE)