Uncertain mileage and
the vehicle choice
Most consumers do not know the exact amount of miles they
will drive per year either in the city or the highway. I am introducing some problems where miles
driven or a component of miles driven is a random variable. The problems require calculation of the
expected amount and variance of gasoline consumed and dollars spent on
gasoline.
I am solving these problems two ways. First using the formulas and properties for
expected value and variance in introductory statistics books. Second, using the Monte Carlo Simulation
methods in Excel.
Question: A person is considering the purchase of two
vehicles  the Lexus RXs 400H and the Saturn Vue Hybrid. Information on the fuel efficiencies for
these two vehicles are presented below.
Fuel Efficiency for Two
SUVs


Lexus RX 400H 4WD

Saturn Vue Hybrid


City

31

27

Highway

27

32

Miles driven in the city for the person buying this vehicle
can be described as 5000+ V where V is a random variable normally distributed
with mean 3000 and standard deviation 200.
Miles driven in the highway for this person can be described
as 4000+W where W is a normally distributed random variable with mean 4000 and
a standard deviation of 1000.
What is the expected value and variance of gasoline
consumption per year for this driver in the Lexus RX and in the Saturn Vue?
What are the expected value and variance of dollars spent on
gas for the two vehicles?
Short Answer: Average fuel efficiency consumed for the two
SUVS is similar. Saturn has the lower
highway mpg and in the scenario presented here the driver has a lot of
uncertainty about his highway mileage.
As a result in this scenario the STD of gas consumption is lower on the
Saturn.
Answer
The solution based on the properties of expected value and
variance.
Expected Gas Consumption = E(Miles)/(mpg)
Var Gas Consumption =Var(Miles/mpg) = (1/mpg^{2}) x
Var(Miles)
Var Gas Expenditures Var (dpg x miles /mpg) = (dpg/mpg)^2 x Var(Miles)
Here dpg is dollars per gallon and mpg is miles per
gallon.
Assume the variable city miles per year is 5000+V where V is
a random variable normally distributed with mean 3000 and standard deviation
500.
Assume the variable highway miles per year is 3000+X miles
per year where X is normally distributed with mean 5000 and standard deviation
800.
Find the mean and standard deviation of gallons of gas
consumed per year for people with miles driven described by these
distributions.
Results:
Calculations based on properties of standard deviation and
variance.
Expected Gasoline
Consumption Two SUVs


mpg Lexus

mpg Saturn

Exp. Miles

Expected Gas Lexus

Expected Gas Saturn

Expected Gas Expendiure
Lexus

Expected Gas Expenditure
Saturn


City

31

27

8000

258

296

$1,032

$1,185

Hway

27

32

8000

296

250

$1,185

$1,000

554

546

$2,217

$2,185

Standard Deviation Gas
Consumption Two SUVs


mpg Lexus

mpg Saturn

Var Miles

STDs Gas Lexus

STD Gas Saturn

STD GAS Expenditure Lexus

STD Gas Expenditure
Saturn


City

31

27

250000

16.1

18.5

$64.5

$74.1

Hway

27

32

640000

29.6

25.0

$118.5

$100.0

33.7

31.1

$134.9

$124.4

Another way to get these results is to use the reverse norm.inv
function in excel to generate vehicle miles driven (city and highway) for a
sample of individuals. The VMT numbers
are used as input for gas consumed and gas expenditures (Again, divide VMT by miles per gallon and
multiply by dollar per gallon.
The Excel formula used to generate vehicle miles traveled in
a city for the Lexus is
=((5000+NORMINV(RAND(),3000,500))/31)
A very good article on the use of random variables in Excel
can be found at the link below.
Many of you have also enjoyed my work on the Toyota Prius.
The links to these posts are presented below.
#Saturn
#Lexus
#fuel efficiency
#Monte Carlo Simulation
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