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Severaladvantagesofthiskindofrepresentationforuncertahvariablescanbeseenimmediately. Atthe simplest level, presenting information in this form acknowledges that the value in question is uncertain. This recognition reduces the temptation to rely solely on deterministic analysis-treating uncertain variables as if they were known with certainty. At the sitme time, developing a density hnction for key uncertain variables lays the foundation for an analysis that makes maximum use of the information the analystdoes haveaboutuncertainvariables,imperfectasitmaybe. Inaddition,variablesthathavebeen described by probability distributions can be used as inputs to formal decision analysis. Figure 2-2 illustrates an alternative presentation of the PDF. Here, the cumulative distribution, which corresponds to the probability density hnction shown in Figure 2-1, is di~p1ayed.l~Figures 2-1 and 2-2 capture the uncertainty in a single variable. However, there will frequently be more than one uncertain variable in an economic evaluation. The (ioint) density function of r such random variables could be representedbyagraphsimilartoFigure2-1butwithr+1dimensionsinsteadof2. Oftentheuncertainty in one variable will be related to the uncertainty in another variable. For example, in the economic evaluation of a dual fuel combustion turbine capable of burning natural gas or oil, the price of the two fuels may be uncertain, and, some would argue, highly correlated. Economic evaluations with such a correlated or dependenti8variable must explicitly consider such correlation. Variability can also be captured by the density and cumulative distribution functions of Figures 2-1 and 2-2. Themagnitudeofthedensityfunctionisafunctionofthefrequencywithwhichthevariabletakes oneachvalueoverthepopulation. Forexample,adensityfunctionportrayingregionalvariationsinthe pricehomeownerspayforelectricitywillhaveitspeakatthemostfrequentlyoccurringprice. Although there are some important distinctions between variability and uncertainty that can require different treatments, the steps described below for economic evaluations under uncertainty can also be applied to economic evaluations with inputs that vary over the population (e.g., variation in the consumer price of compact fluorescent light bulbs). 17Theprobability density function and cumulative distributions contain exactly the same information. One can be derived Erom the other by means of the relationship where F<(-)is the cumulative distribution and f(.) is the probability density function. In the other direction, the relationship between these two functions is simply “A pair of random (uncertain) variables xl, xz are said to be independent if their joint density functions can be represented as the product of their individual (marginal) density functions as follows: Where f(x,, x2) is the joint density function over xi and x, taken together, and f(x,) and f(x,) are the marginal densities of the variables individually. Variables that are not independent are said to be dependent. Theconceptofindependencegeneralizesinastraightforwardwaytosetsofmorethantwo variables. 29PDF Image | A Manual for the Economic Evaluation of Energy Efficiency and Renewable Energy Technologies
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