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Extra info for A Certain Uncertainty: Nature's Random Ways
1 x1 μyÀx 2 Àμ1 μ1 ð1:12:2Þ e ¼ e ðy À x1 Þ! ðx1 Þ! x1 ¼0 y y eÀðμ1 þμ2 Þ X y! eÀðμ1 þμ2 Þ X y x yÀx yÀx ¼ μx1 μ2 ¼ μμ x 1 2 y! y! ðy À xÞ! x¼0 x¼0 ¼ 4 Àμ2 eÀðμ1 þμ2 Þ ðμ1 þ μ2 Þy : y! Hypergeometric functions occur in the solution of second-order differential equations that describe a variety of physical system. e. the Coulomb problem). 12 Poisson moment-generating function 23 The first step is in effect a statement of the sought-for probability by means of Bayes’ theorem. The transition from the first to the second is permitted because the Poisson variates X1 and X2 are assumed independent.
GX ðatÞÞn , eai tXi ¼ i¼1 iid i¼1 ð1:10:1Þ where the third equality is permitted because the random variables are independent. Recall: If A and B are independent, then hABi ¼ hAihBi. The arrow above shows the reduction of gSn ðtÞ in the case of independent identically distributed (iid) random variables all combined with the same coefficient a. n X gXi ðai tÞg00Xi ðai tÞ À g0Xi ðai tÞ2 2 ) σ ¼ a2i σ 2Xi : Sn gX ð ai t Þ 2 i¼1 dln gSn ðtÞ dt ð1:10:3Þ t¼0 i Another special case of particular utility is the equivalence relation for a normal variate X À Á ð1:10:4Þ N μ, σ 2 ¼ μ þ σN ð0, 1Þ, which will be demonstrated later in the chapter.
16 Characteristic function which is recognizable as the Fourier transform of the pdf. In this capacity lies its primary utility, for it permits one to calculate the probability density (or probability function) by an inverse transform ð∞ 1 eÀiXt hX ðtÞdt, ð1:16:3Þ pX ðx Þ ¼ 2π À∞ which cannot always be done so straightforwardly by means of the mgf itself. One can, of course, also calculate moments of a distribution by expansion of hX (t) in a Taylor series about t ¼ 0 to obtain an alternating progression of real and imaginary valued quantities, but I have found little advantage to using it this way when gX(t) is available.
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