工程经济学第11章(英文版)

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Discrete Random Variables
Example: Acme manufacturing has installed a muchneeded new CNC machine. The initial investment in this machine is $180,000 and annual expenses are $12,000. The life of the machine is expected to be 5 years, with a $20,000 market value at that time. Acme’s MARR is 10%. Possible revenues follow the probabilities given below.
End of Year,k Expected Value of Net Cash Flow, Fk SD of Net Cash Flow, Fk
0
1 2 3
-$10,000
$4,000 $4,000 $5,000
$0
$400 $600 $650
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Continuous Random Variables
The expected value (mean, central moment), E(X), and variance (measure of dispersion), V(X), of a random variable X, are
8
Random Variables
Some properties of the mean and variance.
Find the expected PW.
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Continuous Random Variables
Find V(PW) and SD(PW)
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Continuous Random Variables
With our estimates of cash flow variables, and using the normal distribution, we can find the probability of events about the random variable occurring. For instance, in the previous example, what is the probability that the PW of the cash flows is positive? Recall
24
Continuous Random Variables
The standard normal (mean=0 and standard deviation=0) variable, Z, is defined as
For our problem, since our random variable is present worth,
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Discrete Random Variables
Probability tree diagrams can display prospective cash flows, and their respective probabilities that occur in each time period.
Revenue Probability
$35,000
$44,000 $50,000 $60,000
0.1
0.3 0.4 0.2
10
Discrete Random Variables
What is the expected value of Acme’s revenue?
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Discrete Random Variables
Sources of Uncertainty
Decision making is fraught with risk and uncertainty.
• Decisions under risk are those where the decision maker can estimate probabilities of occurrence of particular outcomes. • Decisions under uncertainty are those where estimates of probabilities of the several unknown future states cannot be estimated.
What is the variance of Acme’s revenue?
12
Discrete Random Variables
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Discrete Random Variables
14
Discrete Random Variables
15
Discrete Random Variables
The useful life is uncertain. What are the Expected Value, Variance and SD of the PW?
What is the probability that the PW 0? P(PW 0) = 1 – P(PW<0) P(PW 0) = 1 – (0.1+0.2) P(PW 0) = 0.7
The probability that X takes on any particular value is 0.
6
Random Variables
The cumulative distribution function (CDF) is
which leads to
7
Random Variables
where
4
Random Variables
Some other properties of discrete random variables.
Probability mass function
Cumulative distribution function
5
Random Variables
For continuous random variables…
Apply these concepts to cash flows over time to find the expected PW, and SD of PW, for the expected values below. (Use i=8%.)
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Monte Carlo Simulation
Another way to handle uncertainty is to use Monte Carlo simulation.
• • • • Based on the probability of different outcomes for each random variable, a particular value is randomly generated. The numbers generated for all random variables constitute an instance or realization reflecting a particular outcome. Hundreds or thousands of these instances are generated, and these are examined to assist in decision making. A caution: these will yield long term, average results, and you will be able to see the variation over time. However, your decision may be a one-time decision, so don’t expect the “average” outcome to be your outcome.
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Monte Carlo Simulation
Suppose the useful life for a piece of machinery has a discrete distribution as follows.
Given this distribution. For each trial Select a Random Number and assign the life to each value trial corresponding to the range that is proportional to the respective probabilities.
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Continuous Random Variables
The probability is found by looking up a value in the standard normal table (Appendix E).
So
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Continuous Random Variables
2
Sources of Uncertainty
Four major sources of uncertainty are present in engineering economy studies
1. Possible inaccuracy of cash-flow estimates. 2. The type of business involved in relation to the future health of the economy. 3. The type of physical plant and equipment involved. 4. The length of the study period used in the analysis.