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If X is a discrete random variable taking values x in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as [1] G ( z ) = E ( z X ) = ∑ x = 0 ∞ p ( x ) z x , {\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},} where p {\displaystyle p} is the probability mass ...
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
Points with suffix P are in the Z plane and points with suffix Q are in the Y plane. Therefore, transformations P 1 to Q 1 and P 3 to Q 3 are from the Z Smith chart to the Y Smith chart and transformation Q 2 to P 2 is from the Y Smith chart to the Z Smith chart. The following table shows the steps taken to work through the remaining components ...
For any , the coefficient of /! in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value [] . The cumulant generating function is the logarithm of the moment generating function, namely
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell . Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)} , called the Bell series ...
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
Faà di Bruno's formula gives coefficients of the composition of two formal power series in terms of the coefficients of those two series. Equivalently, it is a formula for the nth derivative of a composite function. Lagrange reversion theorem for another theorem sometimes called the inversion theorem; Formal power series#The Lagrange inversion ...
is the reference distance, usually 1 km (or 1 mile) for a large cell and 1 m to 10 m for a microcell. [1] is the path loss exponent. is a normal (Gaussian) random variable with zero mean, reflecting the attenuation (in decibels) caused by flat fading [citation needed]. In the case of no fading, this variable is 0.