In order to use a piece of information 
derived from a set of
measures x we must have information regarding its likely
variation. If x has been obtained using a measurement system
then we must be able to quantify measurement accuracy. Then we
require a method for propagating likely errors on measurement
through to 
. Assuming knowledge of error covariance this
can be done as follows
The method simply uses the derivative of the function f
as a linear approximation to that function. This is sufficient
provided the expected variation in parameters 
is small compared to the range of linearity of the function.
As an example we can take the Poisson distribution itself which
for large numbers is expected to have a standard
deviation of 
where N is the mean of the distribution.
We will call a sample random variable from this distribution
s. If we now construct a new measure given by
then we can show, using a simplified form of error propagation for one parameter, that the expected variance on t is given by
Thus the distribution of the square-root of a random variable drawn from a Poisson distribution with large mean will be constant. This result will be used to generate the Matusita probability distribution comparison metric.
When the problem not permit algebraic manipulation
in this form due to significant non-linear behaviour in the
range of 
or functional discontinuities then
numerical approaches may be appropriate. These techniques are
often referred to as Monte-Carlo approaches because they
make use of random number generation techniques
to generate sample distributions.