In real life, moments are valuable only to an extent. Just as it is difficult to give an exact account of time, it is equally hard to predict the precise outcome of future moments. Although most discussions of moments focus on discrete or simple systems, this is not the only approach that makes sense in the world. A moment can be thought of as the outcome of a chain of events that has occurred in the background, with each link leading to another and each event contributing towards the larger whole.
Moments, as they occur in the real world, are often referred to as central moments. In statistics, the central moments of a distribution are quantitative estimates based on the mean and standard deviation of the underlying distribution. If the distribution represents population data, then the mean is the central value, and the standard deviation is the variance of random variables taken to be of a uniform probability distribution. The distribution is symmetric, so the central moments are also termed as mean-weights.
The central moments and their distributions can be studied using statistical distributions called log-normal, log-cumulative, or logistic curves. They are often studied in conjunction with the distribution of frequencies, which is a powerful tool for identifying central moments. When the data distribution is a symmetric distribution with a range, then there is some mutual information between the random variables, allowing for reliable forecasts of future values. This mutual information comes from the known frequencies of elements such as the acceleration, velocity, momentum, and acceleration of neighboring elements. It is called the kluster curve.
The probability distribution for moments also follows a log-norm distribution, which plots the normal curve over the data distribution. The probability density function intercepts the log-normal curve at a single point, denoted by p(x), so when plotted on a log-normal curve, it implies that the value of any particular moment, when plotted on a normal curve, will lie along the interval [0, 1]. The likelihood function defines the probability of finding the intercept of a normal curve on the x axis. In cases where the distribution has a non-normal shape, then it can be evaluated over different ranges of x to take into account the inhomial trends common to normal distributions.
Moment distribution and its distributions can also be evaluated using a non-parametric measure called the square root of variance. The square root of variance estimates the variance that exists between any two points in the distribution. The square root of variance directly measures the bias that exists in the normal distribution. The bias is the difference between actual value and expected value, when the data distribution lies on a normal curve.
Moment distribution and its distributions can also be evaluated using a mathematical model called the binomial tree. This model can be used to evaluate central moments, with parameters estimated on a range of inputs, yielding a probability density. This probability density can be compared across different distributions using central moments with parameters estimated on the range of inputs. Moment distribution is a powerful concept that captures both random sampling and non-periodic components of the distribution. Its main advantage over other approaches to quantifying moments is that it is based on an underlying probability structure.