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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Normal distributions ： ウィキペディア英語版 Normal distribution \, e^} | cdf = $\backslash frac12\backslash left(+\; \backslash operatorname\backslash left(\; \backslash frac\backslash ,\backslash operatorname^(2F-1)$ | mean = | median = | mode = | variance = $\backslash sigma^2\backslash ,$ | skewness = 0 | kurtosis = 0 | entropy = $\backslash frac12\; \backslash ln(2\; \backslash pi\; e\; \backslash ,\; \backslash sigma^2)$ | mgf = $\backslash exp\backslash \backslash sigma^2t^2\; \backslash \}$ | char = $\backslash exp\; \backslash \backslash sigma^2\; t^2\; \backslash \}$ | fisher = | conjugate prior = Normal distribution }} In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.〔(''Normal Distribution'' ), Gale Encyclopedia of Psychology〕 The normal distribution is remarkably useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.〔Lyon, A. (2014). (Why are Normal Distributions Normal? ), The British Journal for the Philosophy of Science.〕 Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed. The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as Cauchy's, Student's, and logistic). The terms Gaussian function and Gaussian bell curve are also ambiguous because they sometimes refer to multiples of the normal distribution that cannot be directly interpreted in terms of probabilities. The probability density of the normal distribution is: :$$ f(x \; | \; \mu, \sigma) = \frac \; e^ } Here, $\backslash mu$ is the ''mean'' or ''expectation'' of the distribution (and also its median and mode). The parameter $\backslash sigma$ is its standard deviation with its variance then $\backslash sigma^2$. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate. If $\backslash mu\; =\; 0$ and $\backslash sigma\; =\; 1$, the distribution is called the standard normal distribution or the unit normal distribution denoted by $N(0,1)$ and a random variable with that distribution is a standard normal deviate. The normal distribution is the only absolutely continuous distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance. The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution. The value of the normal distribution is practically zero when the value ''x'' lies more than a few standard deviations away from the mean. Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied. The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. == Definition == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Normal distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.