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Monte Carlo Analysis
Correlation Between Gaussian Input Variables
Correlation Between Gaussian Input Variables
You can specify linear correlations between random variables in Eldo under some conditions:
the variables must have Gaussian distribution and only LOT variations. We review some basic
notions on multivariate probabilities.
Basic Concepts on Multivariate Distributions
A multivariate random variable is a vector of random variables. Notation is made more compact
by writing X = (X 1 , …, X k ) T . Each of the components X i is a random variable in its own right,
but specification of the properties of X as a whole requires information about influence of every
variable on each of the others. Knowledge that one component is large therefore increases the
probability that the other components are large.
Generalizing the single variable case, the joint distribution function of X is defined by:
where x = (x 1 , …, x k ) T . When the X i are continuous random variables, and provided it exists, the
joint density function is given by:
In this case:
The probability density functions of each of the individual X i , termed marginal density
functions, are obtained by integrating out the other components. For example:
is the marginal probability density function of the component X 1 . Similarly:
Eldo® User's Manual, 15.3 437