Question

The water demand of a certain city (in millions of gallons) follows an exponential distribution with a lambda = 1/4. If the capacity of water resources is 10 million gallons.

a. What is the probability of supplies being insufficient on any given day?

b. In the next ten days, what is the probability that we have water shortages in exactly two days?

c. If an inspector arrives, and every day checks for water shortages, how many days on average will he have to maintain his inspection if he stops when he finds two days with water shortages?

d. What is the probability that the total consumption in a week is between 56 and 70 million?

e. What is the probability that the total consumption in a month will be between 200 and 240 million?

Answer 1

a) Suppose X is the water demand of the city

If X is greater than 10 million gallons then the supplies being insufficient on any given day.

So we need to find P( X > 10)

Here Demand (X)  follows an exponential distribution with a lambda = 1/4

Therefore, P( X > 10) is given by

=

therefore p = P( X > 10) = 0.082085

b) In the next ten days, what is the probability that we have water shortages in exactly two days?

here n = 10, p = 0.082085

so by using binomial formula we want to find P( Y = 2)

P ( Y = 2) = =BINOMDIST(2,10,0.082085,0) = 0.152813

here we used " =BINOMDIST(number, trial, n, p, cumulative)" this excel command to find the exact probability of binomial distribution.