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- *PM : moment generating function of the sum of independent random variables, id = 9628 new !-- WP guess : moment generating function of the sum of independent random variables-- Status:
- The authors Kemp and Kemp have called it " Hermite distribution " from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of ( modified ) Hermite polynomials.
- This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is ln ( " x " ).
- Johnson considered the distribution of the logit-transformed variable ln ( " X " / 1 " " X " ), including its moment generating function and approximations for large values of the shape parameters.
- In fact, apparently, Moyal in the mid-1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.
- In the second case, there is no ambiguity provided the problem is " well-posed ", so that no unwarranted assumptions can be made, or have to be made, thereby fixing the appropriate prior probability density function or prior moment generating function ( with variables fixed appropriately ) to be used for the probability itself.
- Finally in 1812, Pierre-Simon Laplace published his " Th�orie analytique des probabilit�s " in which he consolidated and laid down many fundamental results in probability and statistics such as the moment generating function, method of least squares, inductive probability, and hypothesis testing, thus completing the final phase in the development of classical probability.
- The fact that there is a simple expression for the moment generating function implies that simple expressions for all independent random variables that are variance-gamma distributed with the same values of the parameters \ alpha and \ beta, but possibly different values of the other parameters, \ lambda _ 1, \ mu _ 1 and \ lambda _ 2, \ mu _ 2, respectively, then X _ 1 + X _ 2 is variance-gamma distributed with parameters \ alpha, \ beta, \ lambda _ 1 + \ lambda _ 2 and \ mu _ 1 + \ mu _ 2.
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