A homeowner has installed a new energy-efficient furnace. It...

A homeowner has installed a new energy-efficient furnace. It is estimated that over a year the new furnace will reduce energy costs by an amount that can be regarded as a random variable with a mean of $$\$ 200$$ and a standard deviation of $$\$ 60$$. Stating any assumptions you need to make, find the mean and standard deviation of the total energy cost reductions over a period of 5 years.

Key Concepts

Random Variables

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In problems like these, energy cost reductions over a year are treated as random variables because they can vary each year based on different conditions.

Expected Value (Mean) of a Random Variable

The expected value of a random variable is a measure of the center of its distribution, essentially the average outcome if the random process were repeated many times. When calculating the total energy cost reduction over multiple years, the mean of each year's savings can be summed directly to give the total expected reduction.

Standard Deviation and Variance

The standard deviation is a measure of how spread out the values of a random variable are around the mean. Variance, which is the square of the standard deviation, quantifies this spread. When summing independent random variables, their variances add up, and the square root of the total variance gives the overall standard deviation.

Independence and Identically Distributed Assumptions

Assuming the random variables for each year are independent and identically distributed is crucial for summing their means and variances directly. Independence implies that the cost reductions in one year do not affect the reductions in another, allowing for the use of formulas for the mean and variance of sums of random variables.

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