StudentShare
Contact Us
Sign In / Sign Up for FREE
Search
Go to advanced search...
Free

The Utility of the Normal Distribution - Research Paper Example

Cite this document
Summary
The paper "The Utility of the Normal Distribution" states that normal distribution, also known as Gaussian distribution, is usually observed and utilized as the commencing position of modelling numerous natural processes. Numerous physicists slothful and attractive are the theorem known…
Download full paper File format: .doc, available for editing
GRAB THE BEST PAPER95.2% of users find it useful
The Utility of the Normal Distribution
Read Text Preview

Extract of sample "The Utility of the Normal Distribution"

Normal distribution which is also known as Gaussian distribution is normally observed and utilized as the commencing position of modeling numerous natural processes. Numerous physicists slothful and attractive are the theorem known as the central limit theorem. Normal distribution aids in the description of the occurrence of a single random substance and corresponding whole process that act like the Gaussian. Gaussians also aids in minimizing of the entropy for particular energy that mainly pertain to the conserved quadratic quantity energy of the specific formula. Normal distribution was mainly advanced as an approximation to the binomial distribution (Roe, 234-256). The utility of the normal distribution is appreciated to be having amazing property of the physical processes that are random variables, which are utilized to safely approximate the normal distribution. Random variation within the natural processes mostly follows the probability distribution and it is referred as the Gaussian distribution in physics and the bell curve within the social science. The main function of describing the normal distribution has a relatively longer tradition in mathematics and physics. De Moivre utilized it in the approximation of the binomial distribution Laplace utilized it in measurement of errors and Gauss utilized it in the analysis of the astronomical data. Normal distribution and physics used in the computation of errors. Gaussian model is normally represented by the Central Limit Theorem. Central Limit Theorem states that appropriate linear combinations of suitably behaved random variables will be asymptotically shaped thus displaying Gaussian distribution regardless of the underlying distributions of the individual random that are being combined (Benenson et al, 123-167) . Moreover, random variable is normally produced by prevailing linearly combining massive well behaved random variables that are applicable in physics. This is can also be verified by binomial distribution that limit the approximated Gaussian distribution in regard to the discrete random variables. Normal distributions possess numerous convenient properties of random variables with underlying unknown distributions and they all assume the application of normal distribution in physics and astronomy (Roe, 234-256). Approximation normally takes the form of central limit theorem by displaying that the mean of any prevailing variates with any corresponding distribution possess finite mean and variance that normally tends to be the normal distribution. Central Limit Theorem states that the distribution is the sum of a relatively large number of the random variables normally tends to move towards a normal distribution (Benenson et al, 123-167). Moreover, Central Limit Theorem is taken as the rational model of errors thus every step results to small error with the probability distribution. The sum error depicts the final error leads to the normal distribution no matter on the individuals. Expected normal distributions are utilized in determination of errors (Turner, Downing & James, 156-189). A finite number of measurements completely specify the probability distribution of the measurements. Normal or Gaussian distributions is represented by The normalization process of the Gaussian is represented by: Estimation of Mean from a sample Approximation of the most probably value of the mean of the parent population is define by the mean and the corresponding Gaussian distribution. Probability of observing x is: For N measurements of x, the probability of observing that mainly set is product of the P’s: The value of the mean that offers highest probability and corresponds to finding minimum si the sum of the exponent: Thus, the minimum occurs at Error on the approximate of the mean Computations of the mean of the sample as the approximate of the mean of the parent distribution and the underlying mean estimated from the M samples from a parent distribution with the variance Normal distribution is defined as a limiting case of any discrete binomial distribution Pp (n/N) as the sample size N continues to increase in size that is Pp (n/N) becomes normal with mean and variance. µ= Np σ2= Npq The prevailing distribution P(x) is properly normalized since The cumulative distribution function produce probability that variate will postulates value less than or equal to x then the subsequent integral of the underlying normal distribution come to be: Gaussian distribution is broadly utilized statistical distribution within the scientific analysis and other supplementary observational settings (Roe, 234-256). The renowned bell shaped curve is mainly characterized by dual parameters that is the mean (µ) and the standard deviation (σ). The corresponding square of standard deviation gives variance σ2. On the two dimensional plot of distribution, the x-axis parameter ideally represent the independent variable while the corresponding y-axis parameter designate dependent variable that is (y, G(x). The normal distribution is given equation Central Limit Theorem Let x1, x2,x3....xN to designate set of N independent random variates and every X1 possess an arbitrary probability distribution P(x1, x2,x3....xN) with the underlying mean µ1 and variance σ21. The normal form of the variate is designated: Which possess a limiting cumulative distribution function as it approaches normal distribution. In regard to the additional conditions the probability density is displayed to be normal with the mean being zero and variance equivalent to one. The corresponding norm form gives the variate of the nature Which is normally distributed with the µx= µx and corresponding σx= σx/√N The proof of central limit theorem takes into consideration inverse Fourier transform of Px(f). In Fourier analysis possess suitable variance, the normal distribution of the eigenvectors of the Fourier transform operator. Gaussian distribution depicts its own frequency components in relation to the normal distribution to its Fourier transform (Turner, Downing & James, 156-189). Central Limit Theorem under broad range conditions of the probability distribution mainly describes the prevailing sum of the random variables that extends towards Gaussian distribution. The mean and the variance of Gaussian is given by 〈 X 〉=Σµi V (X) =ΣVi=Σσ2 It is not significant for the values of N variables to be identically distributed provided that the subsequent conditions take into considerations rn3=ΣE (∣X i−µi∣3 that ought to be finite for each n and represent the sum of the third central moments. The significant convergence theorems for the underlying identically distributed variables demands that the convergences is monotonic with the N as N escalates the entropy of the prevailing distribution monotonically escalates to approach a normal distribution’s entropy (Roe, 234-256). The third central moment is finite then speed of the convergence, which measured by the difference amidst the actual cumulative distribution and the corresponding normal cumulative distribution at the fixed point. A standard deviation is mainly the measure of variability around the mean and it assumes that the observations of the granted characteristic mainly cluster around the mean within the normal fashion. Moreover, the calculated standard deviation possess extremely convenient property by sixty eight percent of the value that fall in either plus or minus one of the standard deviation from 99.7% values that mainly fall of the prevailing standard deviation from the underlying mean (Roe, 234-256). Global and local boundedness on the structure and the corresponding properties of the state of the natural systems are mainly elucidated by the impact. The boundededness is fundamental condition for keeping the evolution of the natural and artificial systems stable in regard to the long term stability, amount and the rate of energy exchanged in any underlying transition that bounded by the thresholds of stability of the system. The association amidst local and global boundedness and the corresponding stability of the system mainly introduces dual new general properties of the renowned state space and the motion (Benenson et al, 123-167) . The states that are involves includes bounded state and the sequential steps of motions that is ever finite. The immediate consequence of the global boundedness is mainly that of the invariant measure of the state space that represent normal distribution. Therefore, the global boundedness makes the normal distribution ubiquitous for the underlying natural systems. The condition that guarantees the asymptotic stability of the invariant measure that is normal distribution depicts the interrelation amidst the boundedness as an essential condition for the BIS trajectory thus demonstrating the chaotic properties and the corresponding presence of stretching and folding for maintaining the evolution of the trajectory confined within the finite phase volume arbitrary. Work Cited Benenson, Walter, Horst Stöcker, J Harris, & Holger Lutz. Handbook of Physics. New York: Springer, 2006. Print. Roe, Byron P. Probability and Statistics in Experimental Physics. New York, NY: Springer, 2001. Print. Turner, J E, D J. Downing,& James S. Bogard. Statistical Methods in Radiation Physics. Weinheim: Wiley-VCH, 2012. Internet resource. Read More
Cite this document
  • APA
  • MLA
  • CHICAGO
(“You can choose one of these topics: The normal distribution and Research Paper”, n.d.)
You can choose one of these topics: The normal distribution and Research Paper. Retrieved from https://studentshare.org/mathematics/1642233-you-can-choose-one-of-these-topics-the-normal-distribution-and-physics-music-and-fourier-series-the-volume-of-the-unit-ball-in-n-dimensions-geometric-constructions-in-number-theory-applications-of-the-divergence-of-the-harmonic-series-elimination-an
(You Can Choose One of These Topics: The Normal Distribution and Research Paper)
You Can Choose One of These Topics: The Normal Distribution and Research Paper. https://studentshare.org/mathematics/1642233-you-can-choose-one-of-these-topics-the-normal-distribution-and-physics-music-and-fourier-series-the-volume-of-the-unit-ball-in-n-dimensions-geometric-constructions-in-number-theory-applications-of-the-divergence-of-the-harmonic-series-elimination-an.
“You Can Choose One of These Topics: The Normal Distribution and Research Paper”, n.d. https://studentshare.org/mathematics/1642233-you-can-choose-one-of-these-topics-the-normal-distribution-and-physics-music-and-fourier-series-the-volume-of-the-unit-ball-in-n-dimensions-geometric-constructions-in-number-theory-applications-of-the-divergence-of-the-harmonic-series-elimination-an.
  • Cited: 0 times

CHECK THESE SAMPLES OF The Utility of the Normal Distribution

Empirical Evaluation of Value at Risk Model Using the Lusaka Stock Exchange

aR was computed using the standard normal distribution, and other different methodologies of taking into account the non-normality of the returns (the Cornish-Fisher approximation, the modeling of the empirical distribution of the standardized returns and the Extreme Value Theory approach)....
86 Pages (21500 words) Dissertation

Economics: Income Distribution and Poverty

"Economics: Income distribution and Poverty" paper argues that the truly creative minds, which are now too often drowned out in the din of cash registers and bottom lines, will find society's climate somewhat more hospitable when the pressure of money has been reduced.... .... ... ...
10 Pages (2500 words) Essay

Significance of Normal Distribution

the normal distribution can be effectively used to describe with accuracy, any factor or variable which tends to clump or agglomerate surrounding the mean.... In addition, the normal distribution is so significant due to the fact that it is so easy to prepare and work with it.... In the paper 'Significance of normal distribution,' the author focuses on a mathematical model of the frequency distribution of a set of variable data.... In a normal distribution, the model's characteristics are exactly determined....
1 Pages (250 words) Essay

How Does the Distributional Role of the State Conflict with the Objective of Economic Efficiency

This paper outlines economic efficiency, poverty, distribution of income, the role of the state in the redistribution of income.... The market system does not guarantee that everyone will have the same opportunity to accumulate wealth and once an inequality in the distribution of wealth arises it tends to be self-perpetuating because wealth can be inherited.... The distribution of income can be examined in two main ways.... One is by examining the distribution of income between the factors of production....
8 Pages (2000 words) Coursework

Z-Scores and Normal Distribution

The paper entitled 'Z-Scores and normal distribution' presents the information and links which shared to expound readers' understanding of Z-Scores and normal distribution are beneficial for researchers using statistics as methods of interpreting results.... One realized that the effective applications of statistical tools in contemporary research need a more in-depth understanding of statistical concepts such as the standard normal distribution and the standard score....
1 Pages (250 words) Essay

Vary Framework and Its Utility in Risk Management

This coursework "Vary Framework and Its utility in Risk Management" focuses on risk as the key ingredient for profit generation within whatever market-sensitive activity.... An investor's viewpoint of risk is it is all about losing money.... VaR is founded on the same common since fact....
12 Pages (3000 words) Coursework

Outliers in Statistical Analysis

The probability of an outlier to occur in distribution is always there indicating that either there is a measurement error or that the population distribution is heavy-tailed.... The reason for this can be a systematic error that is incidental of challenges in the theory that created the assumption family of distribution or some observations are distant from the mean of data collected.... n cases where the distribution of data is normal, the 3 sigma rule implies that an estimate of one in twenty-two observations will vary by two times the standard deviation or more times from the mean....
8 Pages (2000 words) Coursework

Libertarianism as the Political Philosophy

This paper ''Libertarianism as the Political Philosophy '' tells that In general, libertarianism is the political philosophy that upholds the rights of individuals to liberty.... To keep, attain and interchange their holdings.... It also considers the safeguarding of individual rights as its primary role for the state....
9 Pages (2250 words) Essay
sponsored ads
We use cookies to create the best experience for you. Keep on browsing if you are OK with that, or find out how to manage cookies.
Contact Us