Risk Analysis and Uncertainty in Flood Damage Reduction Studies (2000)
Chapter: Appendix D: Functions and Random Variables
Appendix D
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
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.
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.
