The daily consumption of water in millions of gallons in a c

The daily consumption of water in millions of gallons in a certain city can be treated as a random variable having a gamma distribution with alpha =3 and beta =3. If the water tower of this city has a daily capacity of 15 million gallons find the probability that this water supply will be inadequate on any given day. the probability that this water supply will be adequate on any given day if the daily capacity is changed to 16, 17, 18, 19, and 20 million gallons. You must show the Excel/Matlab programming statements for your answers.

Solution

a) given water tower of the city has a daily capacity of 15 million gallons

the probability that this water supply will be inadequate on the given day i.e., supply is less than requirement is P(x<15)

>> fun = @(x) (1/9)*x.*exp(-x/3);
>> integral(fun,15,Inf)

ans =

0.0404

b) if the daily capacity is increased to 16 probability of adequate is given by P(x>16)

the following matlab comands are used for the calculation of P(x<16)

fun = @(x) (1/9)*x.*exp(-x/3);
>> integral(fun,0,16)

ans =

0.9694

therefore 1-p(x<16)=1-0.9694=0.0306

it changed to 17

integral(fun,0,17)

ans =

0.9769

1-p(x<17)=1-0.9769=0.0231

if capacity is changed to 18

fun = @(x) (1/9)*x.*exp(-x/3);
>> integral(fun,0,18)

ans =

0.9826

1-p(x<18)=1-0.9826=0.0174

if capacity is changed to 19

fun = @(x) (1/9)*x.*exp(-x/3);
>> integral(fun,0,19)

ans =

0.9870

1-p(x<19)=1-0.9870=0.0130

if capacity is changed to 20

fun = @(x) (1/9)*x.*exp(-x/3);
>> integral(fun,0,20)

ans =

0.9902

1-p(x<20)=1-0.9902=0.0098