The average domestic water consumption is 58m^3. The government is proposing to

Question

The average domestic water consumption is 58m^3. The government is proposing to charge customers who have excessive water usage. The government proposes that excessive water usage is any usage that exceeds 1.4 times the average.

(a) Use Markov’s inequality to upper bound the probability of a customer being an excessive user.

(b) What level should the government set the excessive usage threshold to ensure that at most 20% of users are be considered to be excessive users?

(c) Suppose that water usage is modelled using a Pareto distribution with SD = 3 and mean= 136

i. Compute the probability that a customer is an excessive user.

ii. What is the expected value and standard deviation of the water usage per customer.

iii. What distribution is the average water usage of a large number of customers?

Explanation / Answer

a) Markov's inequality says that P( X>=a) <= E(x)/a

here a = 1.4 * 58 = 81.2 ,

So the uppur bound is P (X >= 81.2 ) = 58/81.2 = 0.7143

b) let the level be x times the average usage, therefore using markov's inequality

E(x)/a = 0.2 ; 0.2 = 1/x or x = 5

Hence the govt should set excessive usage at 5 times the average water consumption so that at most 20% of users are considered to be excessive users

c) Not familiar with Pareto distribution also lambda and a are not mentioned

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