Success Probability Varying with the Number of Trials and the Number of Successes
Ch. A. Charalambides · 2016
This chapter focuses on distribution of the number of successes in a given number of trials in a Pólya urn model. It obtains probability function, the q-factorial moments, and the usual factorial moments of the q-Pólya distribution, q-hypergeometric distribution, inverse q-Pólya distribution and inverse q-hypergeometric distribution. The q-hypergeometric distribution may be obtained as the conditional distribution of a q-binomial distribution of the first kind, given its sum with another q-binomial distribution of the first kind independent of it. The negative q-hypergeometric distribution may be obtained as the conditional distribution of a negative q-binomial distribution of the second kind, given its sum with another negative q-binomial distribution of the second kind independent of it. The mixture distribution, with mixed a q-Stirling distribution of the second kind and mixing a q-Stirling distribution of the first kind, is a generalized q-factorial coefficient distribution.