# Risk Analysis and Uncertainty in Flood Damage Reduction Studies(2000)

## Chapter: Appendix D: Functions and Random Variables

« Previous: Appendix C: Economic and Environmental Principles for Water Related Land Resources Implementation Studies
Page 196
Suggested Citation:"Appendix D: Functions and Random Variables." National Research Council. 2000. Risk Analysis and Uncertainty in Flood Damage Reduction Studies. Washington, DC: The National Academies Press. doi: 10.17226/9971.
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## Functions of Random Variables

A random variable X is a variable whose probability of taking on a particular value x in an infinitesimal range is described by a probability density function, f(x). The mean or expected value of X is given by

in which the product f(x)dx is the probability of x occurring in an interval [x, x + dx]. The variance, σ2x, is similarly

When Monte Carlo simulation of a random variable is carried out, a set of n independent values is generated to yield a set of replicates {x1, x2, . . . , xn}, from which the mean is estimated as

The weight, 1/n, implies each value is as likely as any other. Equation 3 represents the process actually used in the Corps's risk analysis procedure, in that the weight, 1/n, approximates the theoretical probability, f(x)dx, and the summation in Equation 3 replaces the integral in

Page 197
Suggested Citation:"Appendix D: Functions and Random Variables." National Research Council. 2000. Risk Analysis and Uncertainty in Flood Damage Reduction Studies. Washington, DC: The National Academies Press. doi: 10.17226/9971.
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Equation 1.

When a sum Z of two random variables, X and Y, is required, the process is more complex. For two variables, x and y, the corresponding z is

z = x + y, (4)

and the expected value of Z is the sum of the expected values of X and Y:

µz= µx+ µy.(5)

However, the variance of Z is

σ2z= σ2x+ σ2y+ 2ρxyσxσy, (6)

where ρxy is the correlation coefficient of x and y (−1 ≤ ρxy ≤ 1). The correlation coefficient introduces a new element into the picture and represents the degree of association of values of x and y. When the variables are statistically independent, ρxy = 0, and the variance of the sum is simply the sum of the variances. When the variables are positively correlated, the variance of the sum is increased by an amount proportional to the degree of correlation.

Similarly, when the difference, Z, between two random variables, X and Y, is found, the value of the variate z can be found as:

z = x − y (7)

and the expected value as

µz= µx− µy, (8)

while the variance of the difference is given by

σ2z= σ2x+ σ2y − 2ρxyσxσy. (9)

In this case, if the variables are positively correlated, the variance of the difference is diminished by an amount proportional to the degree of correlation.

Page 198
Suggested Citation:"Appendix D: Functions and Random Variables." National Research Council. 2000. Risk Analysis and Uncertainty in Flood Damage Reduction Studies. Washington, DC: The National Academies Press. doi: 10.17226/9971.
×

The significance of all these definitions is that Monte Carlo simulation works at the level of replicates, or individually generated values of variables x and y. At that level, the normal rules of arithmetic for sums and differences apply, as specified by equations 4 and 7, and they can also be applied to the expected means of those variables, as given by Equations 5 and 8. However, the variability of a sum or difference of random variables depends in part on the variability in the individual variables and also on the degree of correlation or interdependence between the variables. Properly quantifying variability in a problem involving the interaction of several random variables requires an understanding and a correct representation of their interdependence or correlation.

Page 196
Suggested Citation:"Appendix D: Functions and Random Variables." National Research Council. 2000. Risk Analysis and Uncertainty in Flood Damage Reduction Studies. Washington, DC: The National Academies Press. doi: 10.17226/9971.
×
Page 197
Suggested Citation:"Appendix D: Functions and Random Variables." National Research Council. 2000. Risk Analysis and Uncertainty in Flood Damage Reduction Studies. Washington, DC: The National Academies Press. doi: 10.17226/9971.
×
Page 198
Suggested Citation:"Appendix D: Functions and Random Variables." National Research Council. 2000. Risk Analysis and Uncertainty in Flood Damage Reduction Studies. Washington, DC: The National Academies Press. doi: 10.17226/9971.
×
Next: Appendix E: Biographical Information »
Risk Analysis and Uncertainty in Flood Damage Reduction Studies Get This Book
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Reducing flood damage is a complex task that requires multidisciplinary understanding of the earth sciences and civil engineering. In addressing this task the U.S. Army Corps of Engineers employs its expertise in hydrology, hydraulics, and geotechnical and structural engineering. Dams, levees, and other river-training works must be sized to local conditions; geotechnical theories and applications help ensure that structures will safely withstand potential hydraulic and seismic forces; and economic considerations must be balanced to ensure that reductions in flood damages are proportionate with project costs and associated impacts on social, economic, and environmental values.

A new National Research Council report, Risk Analysis and Uncertainty in Flood Damage Reduction Studies, reviews the Corps of Engineers' risk-based techniques in its flood damage reduction studies and makes recommendations for improving these techniques. Areas in which the Corps has made good progress are noted, and several steps that could improve the Corps' risk-based techniques in engineering and economics applications for flood damage reduction are identified. The report also includes recommendations for improving the federal levee certification program, for broadening the scope of flood damage reduction planning, and for improving communication of risk-based concepts.

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