Lecture 6 Moment-generating functions

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Lecture6: Moment-generating functions1of11Course:Mathematical StatisticsTerm:Fall2017Instructor:Gordan itkovi cLecture6Moment-generating and first propertiesWe use many different functions to describe probability distribution (pdfs,pmfs, cdfs, quantile functions , survival functions , hazard functions , etc.) Moment-generating functions are just another way of describing distribu-tions, but they do require getting used as they lack the intuitive appeal ofpdfs or function (mgf)of the (dis-tribution of the) random variableYis the functionmYof a real param-etertdefined bymY(t) =E[etY],for allt Rfor which the expectationE[etY]is well is hard to give a direct intuition behind this definition, or to explain atwhy it is useful, at this point. It is related to the notions of Fourier transformand generating functions . It will be only through examples in this and laterlectures that a deeper understanding will first order of business is to compute the mgf for some of the more im-portant (named) random variables.

Sep 25, 2019 · Lecture 6: Moment-generating functions 6 of 11 coefficients are related to the moments of Y in the following way: mY(t) = ¥ å k=0 mk k! t k, (6.3.1) where m k = E[Yk] is the k-th moment of Y. A fully rigorous argument of this proposition is beyond the scope of these notes, but we can see why it works is we do the following formal computation

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