By James E. Gentle

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Gentle 32 1 Probability and Distribution Theory Cumulants Another useful sequence of constants for describing the distribution of a scalar random variables are called cumulants. 58) if it exists. Cumulants, except for the first, are also invariant to change in the first moment. 58). 59) µ3 = κ3 + 3κ2 κ1 + κ31 . 58) serves to define the cumulant-generating function. 30 (cumulant-generating function) For the random variable X with moment-generating function ψ(t) the cumulantgenerating function is K(t) = log(ψ(t)).

D. as the two-parameter exponential with parameter (α, θ). ) Let Y = X(1) as before and let Y1 = (Xi − X(1) ). We want to find the distribution of Y1 . Note that Y1 = (Xi − X(1) ) = Xi − nX(1) = (X(i) − X(1) ). Now, for k = 2, . . , n, let Yk = (n − k + 1)(X(k) − X(k−1)). d. as exponential with parameters 0, and θ, and are independent of X(i) . We have independence because the resulting joint density function factorizes. d. exponentials with parameters 0, and θ multiplied by θ is a gamma with parameters n − 2 and 1.

66) i=0 We see that µ[1] = µ1 = µ1 . The r th central factorial moment, denoted µ[r] is the r th factorial moment about µ. We see immediately that the factorial-moment-generating function is the same as the probability-generating function evaluated at t + 1: A Companion for Mathematical Statistics c 2010 James E. Gentle 34 1 Probability and Distribution Theory P (t + 1) = ∞ pj+1 (t + 1)j j=0 = = = ∞ j pj+1 j=0 i=1 ∞ i ∞ i=0 ∞ i=0 t i! j=0 j i t i (pj+1 j(j − 1) · · · (j − i + 1)) ti µ . i! t. the distribution of X.

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A Companion for Mathematical Statistics by James E. Gentle

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