Three isomorphic vector spaces—II

Three isomorphic vector spaces and are defined. The interplay of these vector spaces leads to easy proofs for multinomial identities. Using automorphism each multinomial identity is recast into (k + 1)! − 1 more identities. Distributions arising out of some stopping rules in drawing balls of k colors from an urn with and without replacement are connected so that one can easily go from one to the other. Identities involving joint cdf’s of order statistics are generalized to those coming from arbitrary multivariate distributions. Discrete distribution similar to multinomial is defined, where all the calculations can be done on usual multinomial distribution and transferred to the general.