Question

Each year the water of a stream, river or lake peaks at some level, which may cause flooding of surrounding lands. In any given year, a flood will occur in a river when the maximum annual river stage exceeds 80 feet. a) If the probability that the maximum annual river stage exceeds 80 feet is 0.02, find the probability that the first flood will occur for the first time on the tenth year after this year. b) Suppose that the maximum annual river stage follows an exponential distribution with mean 25 feet. Find the return period τ of a flood.

Answer 1

(a) Given that the probability of exceeding 80 feet is p=0.02, the only way of exceeding the 80 feet for the first time in the tenth year

(b) The mean of an exponential distribution is . So, .

The exponential distribution is given as

The cumulative exponential distribution is

So, we first calculate the probability of exceeding 80 feet. This will be given by

So, the probability of a flood in an year is 0.0408. So, the return period

Return period = 24.53 years

Note that this does not mean that a flood will occur exactly once in 24.53 years. This is an estimate of how frequently the flood will occur.