During the rainy season, we estimate that showers, which significantly increase the flow of a certain river, occur according to a Poisson process with rate λ = 4 per day. Every shower, independently from the others, increases the river flow during T days, where T is a random variable having a uniform distribution on the interval [3,9].
(a) Calculate the mean and the variance of the number of showers that significantly increase the flow of the river
(i) six days after the beginning of the rainy season,
(ii) to days after the beginning of the rainy season, where to ≥ 9.
(b) Suppose that every (significant) shower increases the river flow by a quantity X (in m3/s) having an exponential distribution with parameter 1/10, independently from the other showers and from the number of significant showers. Suppose also that there is a risk of flooding when the increase in the river flow reaches the critical threshold of 310 m3/s. Calculate approximately the probability of flooding 10 days after the beginning of the rainy season.