Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability.
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In the same way as the Dirichlet distribution is the conjugate prior for the categorical distribution, the Dirichlet process is the conjugate prior for infinite, nonparametric discrete distributions.
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Note that for a discrete distribution, computing the CDF is not in general too difficult : we simply add up the individual probabilities for the various points of the distribution.
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A family of distributions is said to belong to a vector exponential family if the probability density function ( or probability mass function, for discrete distributions ) can be written as
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Because the distribution of a continuous latent variable can be approximated by a discrete distribution, the distinction between continuous and discrete variables turns out not to be fundamental at all.
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However, there exists a dynamic programming algorithm that produces a solution for any discrete distribution and also an approximate solution, for any probability distribution, with any desired accuracy.
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It is essentially a mixture of a discrete distribution ( constant 0 ) and a continuous distribution ( a truncated Gaussian distribution with interval ( 0, \ infty ) ).
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According to the de Moivre Laplace theorem, as " n " grows large, the shape of the discrete distribution converges to the continuous Gaussian curve of the normal distribution.
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This can be accomplished by generating a discrete distribution having the same first and second ( and possibly higher ) moments, where each point in the discrete approximation can be directly transformed.
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As a result, other suggestions for measuring market risk is Conditional Value-at-Risk ( CVaR ) that is coherent for general loss distributions, including discrete distributions and is sub-additive.
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