We have two vectors and that represent probability mass functions (pmf’s). Any pmf must satisfy the following two properties: where represents the probability that the discrete random variable takes the value , i.e., and .
The Euclidean distance () between the two pmf’s and can be expressed as
This can be expanded to
Because of the definition in ()
In (), as due to the second equation in (), , therefore
Hence, as by definition as a distance metric, the Euclidean distance between two pmf vectors is always bounded within