정규 분포

수학노트
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개요

  • 고교 과정의 통계에서는 정규분포의 기본적인 성질과 정규분포표 읽는 방법을 배움.
  • 평균이 \(\mu\), 표준편차가 \(\sigma\)인 정규분포의 \(N(\mu,\sigma^2)\)의 확률밀도함수, 즉 가우시안은 다음과 같음이 알려져 있음.\[\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\]
  • 아래에서는 이 확률밀도함수가 어떻게 해서 얻어지는가를 보임.(기본적으로는 가우스의 증명)
  • 가우시안의 형태를 얻는 또다른 방법으로 드무아브르-라플라스 중심극한정리 를 참조.


'오차의 법칙'을 통한 가우시안의 유도

  • 오차 = 관측하려는 실제값 - 관측에서 얻어지는 값
  • 오차의 분포를 기술하는 확률밀도함수 \(\Phi\)는 다음과 같은 성질을 만족시켜야 함. 1) \(\Phi(x)=\Phi(-x)\) 2)작은 오차가 큰 오차보다 더 나타날 확률이 커야한다. 그리고 매우 큰 오차는 나타날 확률이 매우 작아야 한다. 3) \(\int_{-\infty}^{\infty} \Phi(x)\,dx=1\) 4) 관측하려는 실제값이 \(\mu\) 이고, n 번의 관측을 통해 \(x_ 1, x_ 2, \cdots, x_n\) 을 얻을 확률 \(\Phi(\mu-x_ 1)\Phi(\mu-x_ 2)\cdots\Phi(\mu-x_n)\)의 최대값은 \(\mu=\frac{x_ 1+x_ 2+ \cdots+ x_n}{n}\)에서 얻어진다.
  • 4번 조건을 가우스의 산술평균의 법칙이라 부르며, 관측에 있어 실제값이 될 개연성이 가장 높은 값은 관측된 값들의 산술평균이라는 가정을 하는 것임.


정리 (가우스)

이 조건들을 만족시키는 확률밀도함수는 \(\Phi(x)=\frac{h}{\sqrt{\pi}}e^{-h^2x^2}\) 형태로 주어진다. 여기서 \(h\)는 확률의 정확도와 관련된 값임. (실제로는 표준편차와 연관되는 값)


증명

\(n=3\)인 경우에 4번 조건을 만족시키는 함수를 찾아보자.

\(\Phi(x-x_ 1)\Phi(x-x_ 2)\Phi(x-x_ 3)\)의 최대값은 \(x=\frac{x_ 1+x_ 2+ x_ 3}{3}\) 에서 얻어진다.

따라서 \(\ln \Phi(x-x_ 1)\Phi(x-x_ 2)\Phi(x-x_ 3)\) 의 최대값도 \(x=\frac{x_ 1+x_ 2+ x_ 3}{3}\) 에서 얻어진다.

미분적분학의 결과에 의해, \(x=\frac{x_ 1+x_ 2+ x_ 3}{3}\) 이면, \(\frac{\Phi'(x-x_ 1)}{\Phi(x-x_ 1)}+\frac{\Phi'(x-x_ 2)}{\Phi(x-x_ 2)}+\frac{\Phi'(x-x_ 3)}{\Phi(x-x_ 3)}=0\) 이어야 한다.

\(F(x)=\frac{\Phi'(x)}{\Phi(x)}\) 으로 두자.

\(x+y+z=0\) 이면, \(F(x)+F(y)+F(z)=0\) 이어야 한다.

1번 조건에 의해, \(F\) 는 기함수이다.

따라서 모든 \(x,y\) 에 의해서, \(F(x+y)=F(x)+F(y)\) 가 성립한다. 그러므로 \(F(x)=Ax\) 형태로 쓸수 있다.

이제 적당한 상수 \(B, h\) 에 의해 \(\Phi(x)=Be^{-h^2x^2}\) 꼴로 쓸 수 있다.

모든 \(n\)에 대하여 4번조건이 만족됨은 쉽게 확인할 수 있다. (증명끝)


역사

  • 중심극한정리는 여러 과정을 거쳐 발전
  • 이항분포의 중심극한 정리
    • 라플라스의 19세기 초기 버전

확률변수 X가 이항분포 B(n,p)를 따를 때, n이 충분히 크면 X의 분포는 근사적으로 정규분포 N(np,npq)를 따른다


재미있는 사실

  • 정규분포와 중심극한정리에 대한 이해는 교양인이 알아야 할 수학 주제의 하나
  • Galton's quincunx
  • 예전 독일 마르크화에는 가우스의 발견을 기려 정규분포곡선이 새겨짐1950958-Gauss-detail2.jpg



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말뭉치

  1. Normal distribution, also called Gaussian distribution, the most common distribution function for independent, randomly generated variables.[1]
  2. Read More on This Topic statistics: The normal distribution The most widely used continuous probability distribution in statistics is the normal probability distribution.[1]
  3. normal distribution , sometimes called the bell curve, is a distribution that occurs naturally in many situations.[2]
  4. For example, the bell curve is seen in tests like the SAT and GRE.[2]
  5. A smaller standard deviation indicates that the data is tightly clustered around the mean; the normal distribution will be taller.[2]
  6. If the data is evenly distributed, you may come up with a bell curve.[2]
  7. Everything we do, or almost everything we do in inferential statistics, which is essentially making inferences based on data points, is to some degree based on the normal distribution.[3]
  8. And so what I want to do in this video and in this spreadsheet is to essentially give you as deep an understanding of the normal distribution as possible.[3]
  9. And it actually turns out, for the normal distribution, this isn't an easy thing to evaluate analytically.[3]
  10. and if you were to take the sum of them, as you approach an infinite number of flips, you approach the normal distribution.[3]
  11. You can see a normal distribution being created by random chance![4]
  12. From the big bell curve above we see that 0.1% are less.[4]
  13. Use the Standard Normal Distribution Table when you want more accurate values.[4]
  14. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.[5]
  15. The normal distribution model is motivated by the Central Limit Theorem.[5]
  16. Normal distribution is sometimes confused with symmetrical distribution.[5]
  17. The skewness and kurtosis coefficients measure how different a given distribution is from a normal distribution.[5]
  18. The case where μ = 0 and σ = 1 is called the standard normal distribution.[6]
  19. The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena.[7]
  20. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution.[7]
  21. The normal distribution is a probability function that describes how the values of a variable are distributed.[7]
  22. As with any probability distribution, the parameters for the normal distribution define its shape and probabilities entirely.[7]
  23. If a dataset follows a normal distribution, then about 68% of the observations will fall within of the mean , which in this case is with the interval (-1,1).[8]
  24. Although it may appear as if a normal distribution does not include any values beyond a certain interval, the density is actually positive for all values, .[8]
  25. The standardized values in the second column and the corresponding normal quantile scores are very similar, indicating that the temperature data seem to fit a normal distribution.[8]
  26. Let us find the mean and variance of the standard normal distribution.[9]
  27. To find the CDF of the standard normal distribution, we need to integrate the PDF function.[9]
  28. Most of the continuous data values in a normal distribution tend to cluster around the mean, and the further a value is from the mean, the less likely it is to occur.[10]
  29. The normal distribution is often called the bell curve because the graph of its probability density looks like a bell.[10]
  30. The normal distribution is the most important probability distribution in statistics because many continuous data in nature and psychology displays this bell-shaped curve when compiled and graphed.[10]
  31. Converting the raw scores of a normal distribution to z-scores We can standardized the values (raw scores) of a normal distribution by converting them into z-scores.[10]
  32. The diagram above shows the bell shaped curve of a normal (Gaussian) distribution superimposed on a histogram of a sample from a normal distribution.[11]
  33. The tail area of the normal distribution is evaluated to 15 decimal places of accuracy using the complement of the error function (Abramowitz and Stegun, 1964; Johnson and Kotz, 1970).[11]
  34. This guide will show you how to calculate the probability (area under the curve) of a standard normal distribution.[12]
  35. It will first show you how to interpret a Standard Normal Distribution Table.[12]
  36. As explained above, the standard normal distribution table only provides the probability for values less than a positive z-value (i.e., z-values on the right-hand side of the mean).[12]
  37. We start by remembering that the standard normal distribution has a total area (probability) equal to 1 and it is also symmetrical about the mean.[12]
  38. The "normal distribution" is the most commonly used distribution in statistics.[13]
  39. To choose the best Box-Cox transformation—the one that best approximates a normal distribution - Box and Cox suggested using the maximum likelihood method.[13]
  40. The graph of the normal distribution depends on two factors - the mean and the standard deviation.[14]
  41. To find the probability associated with a normal random variable, use a graphing calculator, an online normal distribution calculator, or a normal distribution table.[14]
  42. In the examples below, we illustrate the use of Stat Trek's Normal Distribution Calculator, a free tool available on this site.[14]
  43. The normal distribution calculator solves common statistical problems, based on the normal distribution.[14]
  44. In a normal distribution, data is symmetrically distributed with no skew.[15]
  45. Example: Using the empirical rule in a normal distribution You collect SAT scores from students in a new test preparation course.[15]
  46. The data follows a normal distribution with a mean score (M) of 1150 and a standard deviation (SD) of 150.[15]
  47. A random variable with the standard Normal distribution, commonly denoted by \(Z\), has mean zero and standard deviation one.[16]
  48. The probabilities for any Normal distribution can be reduced to probabilities for the standard Normal distribution, using the device of standardisation.[16]
  49. Crowd size Suppose that crowd size at home games for a particular football club follows a Normal distribution with mean \(26\ 000\) and standard deviation 5000.[16]
  50. The cdf of any Normal distribution can also be found, using technology, without first standardising.[16]
  51. The normal distribution is also useful when sampling data out of a non-normal data set.[17]
  52. A truncated NORMAL distribution can be defined for a variable by setting the desired minimum and/or maximum values for the variable.[18]
  53. For practical purposes, minimum and maximum values that are at least 3 standard deviations away from the mean generate a complete normal distribution.[18]
  54. For a Normal distribution, 99.73 % of all samples, will fall within 3 Standard Deviations of the mean value.[18]
  55. Many other common distributions become like the normal distribution in special cases.[19]
  56. Look at the histograms of lifetimes given in Figure 21.3 and of resistances given in Figure 21.4 and you will see that they resemble the normal distribution.[19]
  57. If you were to get a large group of students to measure the diameter of a washer to the nearest 0.1 mm, then a histogram of the results would give an approximately normal distribution.[19]
  58. However, there is a problem with the normal distribution function in that is not easy to integrate![19]
  59. The normal distribution is also referred to as Gaussian or Gauss distribution.[20]
  60. In a normal distribution graph, the mean defines the location of the peak, and most of the data points are clustered around the mean.[20]
  61. A normal distribution comes with a perfectly symmetrical shape.[20]
  62. The middle point of a normal distribution is the point with the maximum frequency, which means that it possesses the most observations of the variable.[20]
  63. We will get a normal distribution if there is a true answer for the distance, but as we shoot for this distance, since, to err is human, we are likely to miss the target.[21]
  64. We can use the fact that the normal distribution is a probability distribution, and the total area under the curve is 1.[21]
  65. If you use the normal distribution, the probability comes of to be about 0.728668.[21]
  66. The minimum variance unbiased estimator (MVUE) is commonly used to estimate the parameters of the normal distribution.[22]
  67. For an example, see Fit Normal Distribution Object.[22]
  68. The normal distribution is the most well-known distribution and the most frequently used in statistical theory and applications.[23]
  69. Any articles that did not specify the type of distribution or which referred to the normal distribution were likewise excluded.[23]
  70. In stage 2 we eliminated a further 292 abstracts that made no mention of the type of distribution and one which referred to a normal distribution.[23]
  71. Before introducing the normal distribution, we first look at two important concepts: the Central limit theorem, and the concept of independence.[24]
  72. The Central limit theorem plays an important role in the theory of probability and in the derivation of the normal distribution.[24]
  73. As one sees from the above figures, the distribution from these averages quickly takes the shape of the so-called normal distribution.[24]
  74. You might still find yourself having to refer to tables of cumulative area under the normal distribution, instead of using the pnorm() function (for example in a test or exam).[24]
  75. The normal distribution is a continuous, univariate, symmetric, unbounded, unimodal and bell-shaped probability distribution.[25]
  76. Use this to describe a quantity that has a normal normal distribution with the given «mean» and standard deviation «stddev».[25]
  77. Suppose you want to fit a Normal distribution to historical data.[25]
  78. The normal distribution holds an honored role in probability and statistics, mostly because of the central limit theorem, one of the fundamental theorems that forms a bridge between the two subjects.[26]
  79. In addition, as we will see, the normal distribution has many nice mathematical properties.[26]
  80. In the Special Distribution Simulator, select the normal distribution and keep the default settings.[26]
  81. In the special distribution calculator, select the normal distribution and keep the default settings.[26]
  82. Indeed it is so common, that people often know it as the normal curve or normal distribution, shown in Figure \(\PageIndex{1}\).[27]
  83. It is also known as the Gaussian distribution after Frederic Gauss, the first person to formalize its mathematical expression.[27]
  84. The normal distribution model always describes a symmetric, unimodal, bell shaped curve.[27]
  85. Specifically, the normal distribution model can be adjusted using two parameters: mean and standard deviation.[27]
  86. The normal or Gaussian distribution is extremely important in statistics, in part because it shows up all the time in nature.[28]
  87. The standard normal is defined as a normal distribution with μ = 0 and σ = 1.[28]
  88. You probably have explored the normal distribution before.[28]
  89. Below, you can adjust the parameters of the normal distribution and compare it to the standard normal.[28]
  90. For normally distributed vectors, see Multivariate normal distribution .[29]
  91. The simplest case of a normal distribution is known as the standard normal distribution.[29]
  92. Authors differ on which normal distribution should be called the "standard" one.[29]
  93. σ Z + μ {\displaystyle X=\sigma Z+\mu } will have a normal distribution with expected value μ {\displaystyle \mu } and standard deviation σ {\displaystyle \sigma } .[29]
  94. The so-called "standard normal distribution" is given by taking and in a general normal distribution.[30]
  95. This theorem states that the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution.[30]

소스

  1. ↑ 1.0 1.1 normal distribution | Definition, Examples, Graph, & Facts
  2. ↑ 2.0 2.1 2.2 2.3 Normal Distributions (Bell Curve): Definition, Word Problems
  3. ↑ 3.0 3.1 3.2 3.3 Normal distribution (Gaussian distribution) (video)
  4. ↑ 4.0 4.1 4.2 Normal Distribution
  5. ↑ 5.0 5.1 5.2 5.3 Normal Distribution
  6. ↑ 1.3.6.6.1. Normal Distribution
  7. ↑ 7.0 7.1 7.2 7.3 Normal Distribution in Statistics
  8. ↑ 8.0 8.1 8.2 The Normal Distribution
  9. ↑ 9.0 9.1 Normal random variables
  10. ↑ 10.0 10.1 10.2 10.3 Normal Distribution (Bell Curve)
  11. ↑ 11.0 11.1 Normal Distribution and Standard Normal (Gaussian)
  12. ↑ 12.0 12.1 12.2 12.3 How to do Normal Distributions Calculations
  13. ↑ 13.0 13.1 Dietary Assessment Primer
  14. ↑ 14.0 14.1 14.2 14.3 Normal Distribution
  15. ↑ 15.0 15.1 15.2 Examples, Formulas, & Uses
  16. ↑ 16.0 16.1 16.2 16.3 Normal distribution
  17. ↑ Radiology Reference Article
  18. ↑ 18.0 18.1 18.2 Normal Distribution
  19. ↑ 19.0 19.1 19.2 19.3 Normal Distribution - an overview
  20. ↑ 20.0 20.1 20.2 20.3 Overview, Parameters, and Properties
  21. ↑ 21.0 21.1 21.2 The Normal Distribution
  22. ↑ 22.0 22.1 Normal Distribution
  23. ↑ 23.0 23.1 23.2 Non-normal Distributions Commonly Used in Health, Education, and Social Sciences: A Systematic Review
  24. ↑ 24.0 24.1 24.2 24.3 2.8. Normal distribution — Process Improvement using Data
  25. ↑ 25.0 25.1 25.2 Normal distribution
  26. ↑ 26.0 26.1 26.2 26.3 The Normal Distribution
  27. ↑ 27.0 27.1 27.2 27.3 3.1: Normal Distribution
  28. ↑ 28.0 28.1 28.2 28.3 1 Let’s Explore the Normal Distribution
  29. ↑ 29.0 29.1 29.2 29.3 Normal distribution
  30. ↑ 30.0 30.1 Normal Distribution -- from Wolfram MathWorld

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'normal'}, {'LEMMA': 'distribution'}]
  • [{'LOWER': 'gaussian'}, {'LEMMA': 'distribution'}]
  • [{'LOWER': 'bell'}, {'LEMMA': 'curve'}]
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