  We propose the sum and the difference of the normalized velocity of two-joint
systems to describe its long-range interaction. It is found that the
conditional probability distribution function (CPDF) of the normalized velocity
between two-joint systems can be derived. The analytical CPDF needs only six
parameters for arbitrary two-joint systems. Two typical currency exchange
databases, i.e., EUR/USD and GBP/USD, which collect the minutely opening
exchange prices from 1 January 1999 to 31 December 2011, are adopted as
example. By calculating the CPDF in the currency exchange databases, it is
shown that all of the results are well described by our theory. We also use the
analytical CPDF to calculate the entropy of two-joint systems, it is found that
the entropy of two-joint systems is less than the sum of entropy of each system
in the two currency exchange databases. It means that some information of
two-joint systems may overlap. We must important to note that the results
presented here do not need to know the form of interaction of two-joint
systems, and the analytical CPDF is a general expression. It is valid not only
for current exchange systems, but also helpful for investigate other
conditional statistical properties in any two-joint long-range interactions
complex systems.
