Question

Suppose that on average 4 tornados occur in 8 years in a county in Oklahoma. If the occurrence of tornados follows a Poisson process: (a) What is the probability that less than 6 years pass before 2 tornados occur (b) What is the probability that there will be at least one tornado next year? (c) What is the probability that the time between successive tornados is greater than 3 years?

Answer 1

Average number of  tornados occur in a year  = 4/8 = 0.5

Let X be the number of tornados occur in t years.

X ~ Poisson( = 0.5t)

(a)

For t = 6 years,

= 0.5 * 6 = 3

X ~ Poisson( = 3)

Probability that less than 6 years pass before 2 tornados occur = Probability that at least 2 tornados occur in 6 years

= P(X 2)

= 1 - P(X < 2)

= 1 - P(X = 0) - P(X = 1)

= 1 - 0.04978707 - 0.14936121

= 0.8008517

(b)

For t = 1

= 0.5 * 1 = 0.6

X ~ Poisson( = 1)

Probability that there will be at least one tornado next year =  P(X 1)

= 1 - P(X < 1)

= 1 - P(X = 0)

= 1 - 0.6065307

= 0.3934693

(c)

For t = 3 years,

= 0.5 * 3 = 1.5

X ~ Poisson( = 1.5)

Probability that the time between successive tornados is greater than 3 years = Probability that no tornados occurs in next 3 years = P(X = 0)

= 0.2231302