Appendix One to the
Two SUV Problem:
Introductory Comment: A previous post used Excel simulation methods
to provide estimates of the expected value and variance of expenditures on
gasoline for two SUVS.
In the previous problem, models driven on the highway, miles
driven in city and the price of gasoline were assumed to be random variables. Both miles driven random variables were
assumed normally distributed. The
gasoline price variable was assumed uniform.
The assumption of normally distributed highway miles may not
be realistic because on some days people take very long trips on the highway
and on other days people don’t drive on the highway at all.
This problem considers alternative assumptions on highway
models driven and their impact on fuel expenditures. First, I pose a methodological question.
Methodological Question: The previous twoSUV problem assumed that daily
highway models were normally distributed with mean 50 miles per day and
standard deviation of 15 miles per day.
Why would it be inappropriate to change this assumption to N(50,40)?
Answer: We can agree that negative miles traveled is
not possible so the random variable picked to model vehicle miles traveled
should preferably never have a negative outcome. What is the likelihood that a vehicle miles
day N(50,40) is negative?
Use Excel to get
Norm.Dist(0,50,40,True)
= 0.10565.
In the original problem STD=15. Note that
Norm.Dist(0,50,15,TRUE) = 0.0043.
This is pretty close to 0.
We could get rid of all negative vehicle miles traveled days by placing
the NORM.INV function inside a MAX function VMT=MAX(0,NORM.INV). The problem
can also be solved with a different distribution for VMT  perhaps the log
normal distribution.
Alternative
approaches to the SUV simulation problem:
We modify the previous SUV problem by modifying the VMT highway miles
driven assumption.
First, assume that VMT Highway miles driven is distributed
by
VMTHWAY=Max(0,N(50,40).
Second assume that VMT is log normally distributed with mean
log(50) and STD Log(5).
What are the mean and the standard deviation of the Subaru
CrossTrek and the Honda CRV under the new highway VMT assumptions?
KEEP ALL OTHER ASSUMPTIONS FROM THE OLD PROBLEM INTACT.
Solutions with the Alternative Highway VMT assumptions:
Since I am keeping all assumptions from tbe old gas
expenditure model intact I just open the old spreadsheet and modify the highway
VMT model. The spreadsheet automatically
updates.
Note because the spreadsheet automatically updates you will
not be able to exactly replicate my results.
Each simulation run is randomly generated If you recalculate your
spreadsheet you will get new answers.
It is possible in some software to run the simulations many
times and report the average and standard deviation of all simulations. I am not set up to do this.
The new VMT formula for first alternative highway VMT
assumption is:
VMT.HIGHWAY=MAX(0,NORM.INV(RAND(),50,40))
The new
VMT formula for the lognormal assumption is
=LOGNORM.INV(RAND(),LN(50),LN(5))
Simulation Results Under
Alternative Assumptions
for Highway Miles


N(50,15)


Subaru CrossTrek

Honda CRV


Average

$28.80

$36.50

Standard Deviation

$4.10

$5.10

MAX(0,N(50,40)


Subaru CrossTrek

Honda CRV


Average

$29.29

$37.00

Standard Deviation

$5.22

$6.19

LOGNORM(ln(50),ln(5))


Subaru CrossTrek

Honda CRV


Average

$50.0

$59.6

Standard Deviation

$99.6

$108.9

A final note: The Lognormal assumption needs to be tinkered
with. A smaller STD would be more
realistic.
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