4M Planning Operating Costs
Management of a chain of retail stores has the opportunity to lock in prices for electricity and natural gas, the two energy sources used in the stores. A typical store in this chain uses electricity for lighting and air conditioning. In the winter, natural gas supplies heat. Managers at a recent meeting settled on the following estimates of typical annual use of electricity and natural gas by the stores. They estimated the chances for varying levels of use based on their own experiences operating stores and their expectation for the coming long-term weather patterns.
The cost of electricity is roughly
per thousand kilowatt-hours, and the cost of natural gas is about
per thousand cubic feet.
Motivation
(a) Does the company know exactly how much it will spend on energy costs to operate a store in the coming year?
Method
(b) Identify random variables for the amount of electricity that is used (
) and the amount of natural gas that is used (
). What are the marginal probability distributions for these random variables?
(c) Define a third random variable
that combines these two random variables to determine the annual energy operating costs.
(d) We don’t have the joint distribution for
and
. Do you think that it is appropriate to model the two random variables
and
as independent?
Mechanics
(e) Find the expected value and variance of the amount of electricity used (in thousands of kilowatt-hours).
(f) Find the expected value and variance of the amount of natural gas that is used (in thousands of cubic feet).
(g) The correlation between
and
is believed to be
. Using this value, find the mean and variance of
.
(h) How would your answer to part (g) change if the correlation were 0?
Message
(i) Use the properties of
to summarize typical energy operating costs for management of this retail firm.
(j) What assumptions underlie your analysis? Are there any items that you’ve treated like constants that might more accurately be treated as random variables (if you had enough additional information do so)?