Question

2. The mean annual occurrence rate of tornadoes in a region is v - tornadoes/year (a) If the random variable (r.v.) describing the occurrence of torna- does, Xt, can be assumed to follow a Poisson process, find the probabilitv that the next tornado will occur within the next two vears (at least one tornado will occur within the next two vears (b) Using instead the r.v of recurrence time, Ti, that follows an exponential distribution, show that you obta as in part (a) for the probability that the next tornado will occur in the same answer within the next two vears. (c) Continuing with the r.v. of recurrence time, Ti, find the prob- ability of no occurrence of tornadoes within the next 8 years (d) Show that using the Poisson r.v. of tornado occurrence, X,, of part (a), that the probability of no occurrence of tornadoes in the next 8 years agrees with your answer to part (c) (e) Find the return period for the occurrence of tornadoes f) Find the probability of the occurrence of tornadoes during the return period

Answer 1

a)
for 2 years
mean = 0.3 * 2 = 0.6

${\displaystyle P(k{\text{ events in interval}})=e^{-\lambda }{\frac {\lambda ^{k}}{k!}}}$

P(X >= 1) = 1 - P(X = 0)
= 1 - e^(-0.3*2)
= 0.451188

b)
P(T1 < 2)
P(T < t)
= 1 - e^(-lambda t)
= 1 - e^(-0.3 * 2)
= 0.4511883

c)
P(T1 > 8)
=e^(-0.3*8) = 0.09071

d)
for 8 years
mean = 0.8*3= 2.4
P(X = 0) = e^(-2.4) = 0.09071