In probability theory, the distribution of a discrete random variable <math>X is said to be a member of the (a, b, 0) class of distributions if its probability mass function obeys \frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \qquad k = 1, 2, 3, \dots where <math>p_k = P(X = k) (provided <math>a and <math>b exist and are real). There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions.

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  • In probability theory, the distribution of a discrete random variable <math>X is said to be a member of the (a, b, 0) class of distributions if its probability mass function obeys \frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \qquad k = 1, 2, 3, \dots where <math>p_k = P(X = k) (provided <math>a and <math>b exist and are real). There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF). More general distributions can be defined by fixing some initial values of pj and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k: for example the logarithmic distribution and the discrete uniform distribution. The (a, b, 0) class of distributions has important applications in actuarial science in the context of loss models.
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  • June 2009
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  • Talk:Panjer recursion Merger proposal
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  • In probability theory, the distribution of a discrete random variable <math>X is said to be a member of the (a, b, 0) class of distributions if its probability mass function obeys \frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \qquad k = 1, 2, 3, \dots where <math>p_k = P(X = k) (provided <math>a and <math>b exist and are real). There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions.
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  • (a,b,0) class of distributions
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