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Monte Carlo Analysis
Correlation Between Gaussian Input Variables
is the joint marginal density function of (X 1 , X 2 ).
In the special situation where the outcome of one random variable has no effect on the
probability distribution of another, the variables are said to be independent. Formally, the
variables X 1 and X 2 are independent if their joint density function factorizes:
More generally, the influence of one random variable on the probability structure of another is
characterized by the conditional density function:
In the case of independent random variables:
but generally the conditional density function depends also on the value of x 2 .
The notions of expectation and variance apply to each margin in turn to give measures of the
location and dispersion of each marginal component. It is also useful, however, to summarize
the extent of dependence between components; that is the extent to which the components
increase or decrease in harmony. The usual summaries are pairwise. The covariance of the
variable X and Y, having joint density function f X, Y , is defined by:
The covariance coefficient is often re-scaled to obtain a measure on a fixed interval. This leads
to the correlation coefficient, defined by:
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Eldo® User's Manual, 15.3