Engineering Economic Analysis

Expected Value 313

This prQbability distribution function shows that there is a 20% chance of having a negative

PW. It also shows that there is a small 3.3% chance of the PW being $32,590. The three values

used to describe possible annual benefits for the project and the two values for life have been

combined to describe the uncertainty in the project's PW.

Creating a distribution, as in Example 10-6, gives us a much better understanding of

the possible PW values along with their probabilities. The three possibilities for the annual

benefit and the two for the life are representative of the much broader set of possibilities

that really exist. Optimistic, most likely, and pessimistic values are a good way to represent

the uncertainty about a variable.

Similarly the six val1.lesfor the PW represent the much broader set of possibilities. The

20% probability of a negative PW is one measure of risk that we will talk about later in the

chapter.

Some problems, such as Examples 10-3 and 10-4, have so many variables or different

outcomes that constructing the joint probability distribution is arithmetically burdensome.

If the values in Equation 10-1 are treated as a discrete probability distribution function, the

probabilities are 1/6, 2/3, 1/6. With an optimistic, most likely, and pessimistic outcome

for each of 4 variables, there are 34= 81 combinations.In Examples 10-3 and 10-4,

the salvage value has only two distinct values, so there are still 3 x 3 x 3 x 2 = 54

combinations.

When the problem is important enough, the effort to construct the joint probabil-

ity distribution is worthwhile. It gives the analyst and the decision maker a better un-.

derstanding of what may happen. It is also needed to calculate measures of a project's

risk. While spreadsheets can automate the arithmetic, simulation (described at the end

of the chapter) can be a better choice when there are a large number of variables and

combinations.

EXPECTEDVALUE

For any probability distribution we can compute the expected value (EV) or arithmetic

average (mean). To calculate the EV, each outcome is weighted by its probability, and

the results are summed. This is NOT the simple average or unweighted mean. When the

class average on a test is computed, this is an unweighted mean. Each student's test has

the same weight. This simple "average" is the one that is shown by the buttonxon many

calculators.

The expected value is a weighted average, like a student's GPA or grade point average.

For a GPA the grade in each class is weighted by the number of credits. For the expected

value of a probability distribution, the weights are the probabilities..

This is described in Equation 10-5.We saw in Example 10-4,that these expected values

can be used to compute a rate of return. They can also be used to calculate a present worth

as in Example 10-7.

Expected value=OutcomeA xpeA) + OutcomeBxPCB)+... (10-5)

.-- -- ...