Enterprise Risk Management Formula Book

Appendix A.3: Probability Distributions: Distributional mixtures

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A random variable is distributed according to a normal mixture distribution if it is of the form  where  and  are independent random variables,  is a non-negative random variable,  and  is some function of . For example, the t distribution has and  being chi-squared with  degrees of freedom and the standard non-central t distribution has  where  is the non-centrality parameter and  is chi-squared with  degrees of freedom.


A distributional mixture of normal distributions is to be interpreted more generally as any distribution in which the overall random variable is selected with probability  from a (typically finite) number of normal random distributions, the ’th one of which is  for arbitrary constant  and . Any univariate distribution can be approximated arbitrarily accurately with a large enough number of underlying normal random distributions. It is contrasted with a linear combination mixture of normal distributions in which the overall random variable is derived by adding together a linear combination of underlying normal random variables, i.e. .


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