Question

Katrina is a seismologist measuring the incidence of earthquakes in California. From past data, she concluded that on the average, four earthquakes occur in California every year. It is widely believed that the number of earthquakes that occur in California in a year is a Poisson random variable.

(a) What is the probability that more than 3 earthquakes are observed in California in a ran- domly selected year?

(b) The time, measured in years, between earthquakes is an exponential random variable with mean equal to 0.25. An earthquake was just observed today. What is the probability that the waiting time for the next earthquake is more than 1 year?

(c) Katrina is interested in the total time that she has to wait to observe exactly two earthquakes after the earthquake observed in part (b). What is the proba- bility that she will wait for more than 18 months in order to observe exactly two earthquakes after the one observed in part (b)?

Answer 1

a)P(X>3) =1-P(X<=3)=1-(P(X=0)+P(X=1)+P(X=2)+P(X=3))

=1-(e^-4*4⁰/0!+e^-4*4¹/1!+e^-4*4²/2!+e^-4*4³/3!) =1-0.4335 =0.5665

b)

probability that the waiting time for the next earthquake is more than 1 year =P(T>1)=e^-t/ =e^-1/0.25 =e^-4 =0.0183

c)expected earthquake in 18 months =18*4/12=6

proba- bility that she will wait for more than 18 months in order to observe exactly two earthquakes after the one observed in part (b) =P(at most 1 earthquake in 18 months) =P(X<=1)=P(X=0)+P(X=1)=e^-6*6⁰/0!+e^-6*6¹/1! =0.0174