Contents
How are Taylor expansions used in probability theory?
Taylor expansions for the moments of functions of random variables. In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
What is the Taylor expansion of f ( x )?
4.2 Taylor Expansion. The Taylor expansion is the standard technique used to obtain a linear or a quadratic approximation of a function of one variable. Recall that the Taylor expansion of a continuous function f (x) is.
How to prove the Taylor expansion of Y N?
Proof: The Taylor expansion of g(Y n) around Y n= is g(Y n) = g( ) + g0( )(Y n ) + Remainder; where the remainder !0 as Y n! . From the assumption that Y nsatis es the standard CLT, we have Y n! in probability, so it follows that the remainder !0 in probability as well. Rearranging terms, we have p n(g(Y n) g( )) = g0( ) p n(Y n ) + Remainder:
Is the Taylor expansion a matrix or vector function?
(Where ℛ 2 represents all the terms of higher order than 2, and a is a ‘convenient’ value at which to evaluate f ). This technique can be expected to matrix and vector functions.
Which is the first moment of the probability distribution?
The mean value of x is thus the first moment of its distribution, while the fact that the probability distribution is normalized means that the zeroth moment is always 1. The jth central moment about x o, in turn, may be defined as the expectation value of the quantity x minus x o, this quantity to the jth power, i.e. .
Is there a way to determine the probability distribution?
If all of the moments are in hand, one can in principle determine the probability distribution itself. One path to this result involves the distribution’s characteristic function, which can be expressed by Taylor series expansion of the exponential thus yielding an infinite sum of moments:
When is the variance the second central moment?
The variance of x is thus the second central moment of the probability distribution when x o is the mean value or first moment. The first central moment is zero when defined with reference to the mean, so that centered moments may in effect be used to “correct” for a non-zero mean.