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. This technique is often used by statisticians.
Noting that E[X − μX] = 0, the 2nd term disappears. Also E[(X − μX)2] is . Therefore,
where μX and are the mean and variance of X respectively.
It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example,
This is a special case of the delta method. For example,
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… … Wikipedia
Moment (mathematics) — Second moment redirects here. For the technique in probability theory, see Second moment method. See also: Moment (physics) Increasing each of the first four moments in turn while keeping the others constant, for a discrete uniform distribution… … Wikipedia
Delta method — In statistics, the delta method is a method for deriving an approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator. More broadly, the delta… … Wikipedia
List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… … Wikipedia
List of mathematics articles (T) — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie … Wikipedia
Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function … Wikipedia
Noncentral chi-squared distribution — Noncentral chi squared Probability density function Cumulative distribution function parameters … Wikipedia