Question: A tour guide
drives his guest in an SUV. The average
tour is 50 miles of city driving. The
standard deviation of miles driven is 8 miles.
The tour guide can get a tour in the morning and a tour in the
afternoon. The probability of getting a
tour in the morning is independent of getting a tour in the afternoon. The probability of a morning tour is 0.5 and
the probability of an afternoon tour is 0.7.
Each tour is exactly 150 miles driven in a city. The tour can occur in a Honda CRVVEX with
a fuel efficiency of 16mpg city or a Subaru XV Cross trek with a 21 mpg in the
city.
What is the expected gas consumption for the tour operator
with the Honda CRV and for the Subaru Cross Trek?
What is the variance of the daily gas consumption?
Answer:
E(Gas) = E (Miles)/mpg
E(Gas Exp) = cpg * E(miles) / mpg
Here mpg is miles per gallon and cpg is cost per gallon.
Calculate expected miles.
E(Miles) = 0.5 *150 + 0.7 *150 = (75+90)=165.
Plug into the gas and consumed equations for the two cars.
Results summarized here.
Tour Guide Fuel
Consumption for Honda CRV and
Subaru Cross Trek Hybrid


Honda CRV

Subaru Cross Trek


City Fuel Efficiency

16

32

Expected Gas In Gallons

10.31

7.86

Expected Gas Expenditure
@ $4.00

$41.25

$31.43

The variance of miles driven should be
150 x 0.5 x 0.5 + 150 x 0.75 x 0.25=65.625.
Look here for the formula of the variance of the Bernoulli
distribution.
Using the properties of variance we can derive the variance
of gas expenditures.
Var((ccg/mpg) x Miles)
= (cpg/mpg)^{2} x Var(miles)
Variability of Tour Guide
Gas Expenditures


Honda CRV

Subuaru Cross Trek


cpg/mpg squared

0.0625

0.015625

Variance $ Gas

$4.10

$1.03

Standard Deviation$ Gas

$2.03

$1.01

Concluding Remark: The variance of gas consumption and
expenditures on gas consumption is much lower for the tour guide that uses the
Subaru Cross Trek than the Honda CRV.
There is only one source of variance in gas consumption for
the tour guide in this example 
uncertainty the number of clients. In
the real world there is uncertainty about other variables including price of
gas and miles driven per trip. I am planning
future posts that consider other factors affecting the variability of fuel
consumed and expenditures on fuel.
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