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In a sequence of independent Bernoulli trials, by countingmultidimensional lattice paths in order to compute theprobability of a first-passage event, we derive and study ageneralized negative binomial distribution of order k, typeI, which extends to distributions of order k, the generalizednegative binomial distribution of Jain and Consul (1971), andincludes as a special case the negative binomial distribution oforder k, type I, of Philippou et al. (1983). This newdistribution gives rise in the limit to generalized logarithmicand Borel-Tanner distributions and, by compounding, to thegeneralized Pólya distribution of the same order and type.Limiting cases are considered and an application to observed datais presented

international journal of mathematics and mathematical sciences

Generalized distributions of order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$k$"><mml:mi>k</mml:mi></mml:math>associated with success runs in Bernoulli trials

Generalized distributions of orderkassociated with success runs in Bernoulli trials