3. A water reservoir is depleted a: a rate a Polsson of 1.000 s per day. A rairf

Question

3. A water reservoir is depleted a: a rate a Polsson of 1.000 s per day. A rairfall adds an amount of water to the reservoir, distributed with a rate 02 rainfalls per day. A The reservoir is refled by rairfalls which oocur as ad uniformly between 400 and reservoir is 5,000 gallous. alloas. The r a) Find the probability that reservoir be empty after five days. (10 points) (OnSL n i - 20 950 Find the expected water level in the reservoit after Eve days. (10 peints) 2 c) Find the probability that a rainfall will happen within four days. (10 points)

Explanation / Answer

a) The reservoir currently has 5000 gallons and water depletion rate is 1000 gallons per day. For the reservoir to get empty within 5 days, there needs to be no rainfall in the nexy five days.

Prob (no rainfall in 5 days) = Exp(0.2*5) = 0.3679

b) Let X be the number of rainfalls in 5 days, and the water added is (400+1300)X/2 = 850X.

Since E(X) = 0.2*5 = 1

E(850X) = 850*E(X) = 850*1 = 850 gallons.

c) Probability that it rains atleast once in 4 days = 1 - Prob. that it doesn't rain at all = 1 - e(-0.2*4) = 1 - 0.449 = 0.55

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