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Probability generating functions are particularly useful for dealing with functions of independent random variables. For example: If , =,,, is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
An ODE problem can be expanded with the auxiliary variables which make the power series method trivial for an equivalent, larger system. Expanding the ODE problem with auxiliary variables produces the same coefficients (since the power series for a function is unique) at the cost of also calculating the coefficients of auxiliary equations.
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
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 ...
[5] [10] [3] [4] [11] [12] Automatic differentiation is particularly important in the field of machine learning. For example, it allows one to implement backpropagation in a neural network without a manually-computed derivative.
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).
increased by 14.3 million and accounted for 26 percent of the growth” (PEW HISPANIC CENTER: 2005, 4). More than 11,000,000 Hispanic children are currently between the ages of 5 and 17. In terms of education policies that impact Hispanics, several major trends stand out: Low enrollment of Hispanic children in early childhood programs and
The syntax of the Python programming language is the set of rules that defines how a Python program will be written and interpreted (by both the runtime system and by human readers). The Python language has many similarities to Perl, C, and Java. However, there are some definite differences between the languages.