## Wednesday, August 13, 2014

### Uncertain mileage and vehicle choice.

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.

The solution based on the properties of expected value and variance.

Expected Gas Consumption = E(Miles)/(mpg)

Var Gas Consumption =Var(Miles/mpg) = (1/mpg2) 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