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For example, Gaussian distribution is fully decided by the first two moments, i.e., mean and variance. Another example is the generalized Gaussian distribution which has one additional parameter, i.e., shaping parameter p, to control the fourth moment.

So my question is whether there exists a similar distribution that fully decided by all the first four moments?

kawofengche
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  • Any distribution that is a member of the Pearson family is defined by the first 4 moments, so that is a good starting point. If you wish, you can then just exclude those (like the Normal) that are defined by less than 4 moments. – wolfies Jan 29 '16 at 14:38
  • @wolfies, thank you, that is really a good starting point. Besides the Pearson family, there are actually many distribution aiming for this situation. For example the asymmetric generalized gaussian distribution. – kawofengche Feb 09 '16 at 13:54

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