The normalized probability density function (pdf) 
of a random
variable is defined such as:
the form of depends on the nature of the variations of the random
variable. In particular, for a uniform deviate random number whose value is
selected from the range 
with each number in that range having equal
probability of occurence:
Given a random variable 
with a pdf , its mean or
average (
), and its variance which is the square of its standard deviation
are given by:
The above integrals are estimated using Monte Carlo integration by numerical summation as:
where n is the number of samples.
The single variable definitions above can be generalized to
the case of multiple variables by introducing the concept of a joint
probability density function (jpdf):
However, in the presence of more than one random variable, one has to take
into account the relationship between the different distributions. A measure
of this correlation is their covariance integral (
):
where 
is the jpdf of 
and 
.
This again can be estimated by:
Normalizing the covariance integral results in the correlation coefficient of
two distributions 
such as 
:
Using the above definitions, a normal pdf is characterized by its mean
and standard deviation :
and the multi-dimensional normal jpdf can be expressed as:
where 
is a vector of n random normal variates
, with a mean vector
, and 
is the
determinant of the 
covariance matrix 
:
Each diagonal terms of this matrix 
is the variance of the 
th
variate, and each off-diagonal term 
is the covariance between the
th and 
th variates.
Multinormal parameter distribution are commonly assumed in semiconductor
manufacturing. Whereas the Monte Carlo technique can be used in conjunction
with any parameters pdf, in the following section we describe the procedure
used to generate normal and multinormal distribution samples only.
