# Question

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 $100 per thousand kilowatt-hours, and the cost of natural gas is about $10 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 (X) and the amount of natural gas that is used (Y). What are the marginal probability distributions for these random variables?

(c) Define a third random variable T that combines these two random variables to determine the annual energy operating costs.

(d) We don’t have the joint distribution for X and Y. Do you think that it is appropriate to model the two random variables X and Y 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 X and Y is believed to be r = 0.4. Using this value, find the mean and variance of T.

(h) How would your answer to part (g) change if the correlation were 0?

Message

(i) Use the properties of T 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)?

The cost of electricity is roughly $100 per thousand kilowatt-hours, and the cost of natural gas is about $10 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 (X) and the amount of natural gas that is used (Y). What are the marginal probability distributions for these random variables?

(c) Define a third random variable T that combines these two random variables to determine the annual energy operating costs.

(d) We don’t have the joint distribution for X and Y. Do you think that it is appropriate to model the two random variables X and Y 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 X and Y is believed to be r = 0.4. Using this value, find the mean and variance of T.

(h) How would your answer to part (g) change if the correlation were 0?

Message

(i) Use the properties of T 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)?

## Answer to relevant Questions

1. Expression for the variance of Y 2. Probability that the first trial B1 fails. 3. Expression for the mean of X 4. Probability that Y is zero 5. Probability that X is 2 6. Covariance between B1 and B2 7. Expression for the ...If the underlying trials are positively dependent (meaning that a success, say, is more likely followed by another success), would you expect the variance of the total number of successes to be larger or smaller than the ...An insurance company offers two policies that cover the cost of repairs from driving accidents. A policy costing $1,000 annually has a $1,000 deductible (meaning the driver is responsible for paying the first $1,000 in ...A marketing research firm interviewed visitors at a car show. The show featured new cars with various types of environmentally oriented enhancements, such as hybrid engines, alternative types of fuels (such as bio fuels), ...One of these pairings of histogram with normal quantile plot shows data that have been rounded. The other shows the same data without rounding. The rounding in the example is similar to the rounding in SAT scores. (a) Which ...Post your question

0