STAT312 Lecture Notes - Random Variable, Binomial Distribution, Normal Distribution
5 Apr 2014
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The moment generating function of a r. v. is the function ( ) = [ ], provided this exists (i. e. is nite) in an open neighbourhood of 0 (i. e. ], provided this exists (i. e. is nite) in an open neighbourhood of 0 (i. e. for gives the characteristic function, which always exists: it is. Note that solutely) in a neighbourhood of = 0 i radius of convergence for | | ), so that it converges (ab- has a: assume this. Then log we have, by the preceding theorem, Continuing, ( )(0) = [ (i. e. we can di erentiate within the. E. g. p( ) with ( ) = x=0. The cumulants of a distribution are de ned in the expansion of the as the coe cients. Cumulant generating function (c. g. f. ) ( ) = log [ Thus (how?) the poisson distribution has all cumulants = .
