The Moment-Generating Function and Negative Integer Moments

On the sum of and Gaussian random variables

“…Although maybe not of major current interest these expectations are similar to the negative moments of a distribution, see Cressie et al (1981). 6.…”

Ergodic capacity of coherent optical wireless communications in Gamma-Gamma turbulence

“…As a result, when the average SNR~e approaches infinity, the leading term in (21) with p = 0 becomes dominant. Therefore, the asymptotic value of 92,c in large SNR regimes can be directly obtained as asym _ r(a - (3) (a,B) Also when the average SNR Ie approaches infinity, the value of the cutoff SNR~o approaches unity, and consequently, log2(10) approaches zero.…”

“…Finally, applying (10) and (12) to (11), one obtains a series ex-Using a similar approach described in [18], we can show (21) pression involving the cutoff SNR 1'0. For min{a, (3} > 1 such with finite number of terms rapidly converges.…”

“…Due to space expression can be numerically solved along with (6) in order to limitation such analysis is omitted. In (21), wp(x, Y, 10) is compute the cutoff SNR 1'0 for a given average SNR and other defined as channel parameters. However, there is a unique advantage of (11) over (6) when the average SNR is asymptotically large, i.e., I'e -+ 00.…”

C245. Inverse moments of negative- binomial distributions

“…In general, we require an expression 2 2 ) to be integer allows expansion of the integrand by the binomial theorem. In the case v = a (an integer) we obtain This expression was obtained by Cressie et al (1981) in one of their examples and yeilds (9) of Zacks (1980) for v = 2 as well as affording a considerable simplification of Zacks's expression (10). Other examples quoted by Zacks (1980) include expressions for the cases a = v-1 and a=v-2 (v not necessarily integral) which also follow simply from (5) (note that Zacks's (3) follows since E$,,{(v+…”