Distance of understanding (Mathematics)

In mathematics,MomentIs, Loosely speaking, a quantitative measure of the shape of a set of points. the "second moment", for example, is widely used and measures the "width" (in a particle sense) of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it cocould be fit by an ellipsoid. other moments describe other aspects of a distribution such as how the distribution is skewed from its mean, or peaked. the mathematical concept is closely related to the concept of Moment in physics, although moment in physics is often represented somewhat differently. any distribution can be characterized by a number of features (such as the mean, the variance, the skewness, etc .), and the moments of a function^{[1] describe the nature of its distribution.}

The 1st moment is denotedμ1. The first moment of the distribution of the random variableXIs the expectation operator, I. e., the population mean (if the first moment exists ).

In higher orders, the central moments (moments about the mean) are more interesting than the moments about zero.KTh Central Moment, of a real-valued random variable Probability DistributionX, With the expected valueμIs:

The first central moment is thus 0. The zero-th Central Moment,μ0 is one. See also central moment.

Significance of the importance of the moments distance

TheNTh moment of a real-valued continuous functionF(X) Of a Real Variable about a valueCIs

It is possible to define moments for random variables in a more general fashion than moments for real values-see moments in metric spaces. the moment of a function, without further explanation, usually refers to the above expressionC= 0.

Usually, sums T in the special context of the problem of moments, the functionF(X) Will be a probability density function.NTh moment about zero of a probability density functionF(X) Is the expected valueXNAnd is calledRaw momentOrCrude moment.^{[2] The moments about its mean μ are calledCentralMoments; these describe the shape of the function, independently of translation.}

IfFIs a probability density function, then the value of the integral above is calledNTh moment of the probability distribution. More generally, ifFIs a cumulative probability distribution function of any probability distribution, which may not have a density function, thenNTh moment of the probability distribution is given by the Riann-Stieltjes integral

WhereXIs a random variable that has this distribution andEThe expectation operator or mean.

When

Then the moment is said not to exist. IfNTh moment about any point exists, so does (N− 1) th moment, and all lower-order moments, about every point.

Variance variance main article: variance

The second central moment about the mean is the variance, the positive square root of which is the standard deviationσ.

Normalized moments Regular distance

TheNormalized NTh central moment or standardized moment isNTh Central Moment dividedσN; The normalizedNTh Central MomentX= E ((X−μ)N)/σN. These normalized central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale.

Skewness
Main article: skewness

The third central moment is a measure of the lopsidedness of the distribution; any specified Ric distribution will have a third Central Moment, if defined, of zero. the normalized third central moment is called The skewness, often Gamma. A distribution that is skewed to the left (the tail of the distribution is heavier on the left) will have a negative skewness. A distribution that is skewed to the right (the tail of the distribution is heavier on the right), will have a positive skewness.

For distributions that are not too different from the normal distribution, the median will be somewhere near μ−γ σ/6; the mode about μ−γ σ/2.

Kurtosis peak mode
Main article: Kurtosis

The fourth central moment is a measure of whether the distribution is tall and skinny or short and squat, compared to the normal distribution of the same variance. since it is the expectation of a fourth power, the Fourth Central Moment, where defined, is always non-negative; and has t for a point distribution, it is always strictly positive. the fourth Central Moment of a normal distribution is 3σ 4.

The kurtosis Kappa is defined to be the normalized Fourth Central Moment minus 3. (equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance.) Some authorities^{[3][4] Do not subtract three, but it is usually more convenient to have the normal distribution at the origin of coordinates. if a distribution has a peak at the mean and long tails, the fourth moment will be high and the kurtosis positive (leptokurtic); and conversely; thus, bounded distributions tend to have low kurtosis (platykurtic ).}

The kurtosis can be positive without limit, but Kappa must be greater than or equal to gamma 2 −2; equality only holds for Binary distributions. for unbounded skew distributions not too far from normal, Kappa tends to be somewhere in the area of gamma 2 and 2 gamma 2.

The inequality can be proven by considering

WhereT= (X−μ)/σ. This is the expectation of a square, so it is non-negative whateverAIs; on the other hand, it's also a quadratic equation inA. Its discriminant must be non-positive, which gives the required relationship.

Mixed moments

Mixed momentsAre moments involving multiple variables.

Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.

Higher moments high-level distance

High-order momentsAre moments beyond 4th-order moments. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality.^{[Citation needed]}

Cumulants cumulative distance

Main article: Cumulant

The first moment and the second and thirdUnnormalized CentralMoments are additive in the sense that ifXAndYAre independent random variables then

And

And

(These can also hold for variables that satisfy weaker conditions than independence. The first always holds; if the second holds, the variables are called uncorrelated ).

In fact, these are the first three cumulants and all cumulants share this additivity property.

Sample moments Sample distance

The moments of a population can be estimated using the sampleK-Th moment

Applied to a sampleX1,X2 ,...,XNDrawn from the population.

It can be shown that the expected value of the sample moment is equal toK-Th moment of the population, if that moment exists, for any sample sizeN. It is thus an unbiased estimator.

Problem of moments distance

TheProblem of momentsSeeks characterizations of SEQUENCES {μ′N:N= 1, 2, 3,...} That are sequences of moments of some functionsF.

Partial moments part distance/side distance

Partial moments are sometimes referred to as "one-sided moments."NTh order lower and upper partial moments with respect to a reference pointRMay be expressed

Partial moments are normalized by being raised to the power 1/N. The upside potential ratio may be expressed as a ratio of a first-order upper Partial Moment to a normalized second-order Lower Partial Moment.

Distance in the moments in Metric Spaces matrix space

Let (M,D) Be a metric space, and let B (M) Be the Borel σ-algebra onM, The σ-algebra generated byD-Open subsetsM. (For technical reasons, it is also convenient to assume thatMIs a separable space with respect to the metricD.) Let 1 ≤P≤ + ∞.

ThePTh momentOf a measureμOn the measurable space (M, B (M) About a given pointX0 inMIs defined to be

μIs said to haveFinitePTh momentIfPTh momentμAboutX0 is finite for someX0M.

This terminology for measures carries over to random variables in the usual way: if (Ω, Σ,P) Is a probability space andX: Ω →MIs a random variable, thenPTh momentOfXAboutX0MIs defined to be

AndXHasFinitePTh momentIfPTh momentXAboutX0 is finite for someX0M.

 

 

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Distance of understanding (Mathematics)