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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 Half-normal distribution ： ウィキペディア英語版 Half-normal distribution \right) \quad x>0 | cdf = $F(x;\; \backslash sigma)\; =\; \backslash mbox\backslash left(\backslash frac\backslash mbox^(F)$ | mean = $\backslash frac\}$ | median = $\backslash sigma\backslash sqrt\backslash mbox^(1/2)$ | mode = | variance = $\backslash sigma^2\backslash left(1\; -\; \backslash frac\backslash right)$ | skewness = | kurtosis = | entropy = $\backslash frac\; \backslash log\; \backslash left(\; \backslash frac\; \backslash right)\; +\; \backslash frac$ | mgf = | char = | fisher = }} The half-normal distribution is a special case of the folded normal distribution. Let $X$ follow an ordinary normal distribution, $N(0,\backslash sigma^2)$, then $Y=|X|$ follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero. Using the $\backslash sigma$ parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by : $f\_Y(y;\; \backslash sigma)\; =\; \backslash frac\}\backslash exp\; \backslash left(\; -\backslash frac\; \backslash right)\; \backslash quad\; y>0$, Where $E()\; =\; \backslash mu\; =\; \backslash frac\}$. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if $\backslash sigma$ is near zero), obtained by setting $\backslash theta=\backslash frac\}$, the probability density function is given by : $f\_Y(y;\; \backslash theta)\; =\; \backslash frac\backslash exp\; \backslash left(\; -\backslash frac\; \backslash right)\; \backslash quad\; y>0$, where $E()\; =\; \backslash mu\; =\; \backslash frac$. The cumulative distribution function (CDF) is given by : $F\_Y(y;\; \backslash sigma)\; =\; \backslash int\_0^y\; \backslash frac\backslash sqrt\}\; \backslash ,\; \backslash exp\; \backslash left(\; -\backslash frac\; \backslash right)\backslash ,\; dx$ Using the change-of-variables $z\; =\; x/(\backslash sqrt\backslash sigma)$, the CDF can be written as : $F\_Y(y;\; \backslash sigma)\; =\; \backslash frac\backslash sigma)\}\backslash exp\; \backslash left(-z^2\backslash right)dz\; =\; \backslash mbox\backslash left(\backslash frac\backslash right),$ where erf(x) is the error function, a standard function in many mathematical software packages. The quantile function (or inverse CDF) is written: :$Q(F;\backslash sigma)=\backslash sigma\backslash sqrt\backslash ,\backslash mbox^(F)$ where $0\backslash le\; F\; \backslash le\; 1$ and $\backslash mbox^()$ is the inverse error function The expectation is then given by : $E(Y)\; =\; \backslash sigma\; \backslash sqrt,$ The variance is given by : $\backslash operatorname(Y)\; =\; \backslash sigma^2\backslash left(1\; -\; \backslash frac\backslash right).$ Since this is proportional to the variance σ2 of ''X'', σ can be seen as a scale parameter of the new distribution. The entropy of the half-normal distribution is exactly one bit less the entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus, : $H(Y)\; =\; \backslash frac\; \backslash log\; \backslash left(\; \backslash frac\; \backslash right)\; +\; \backslash frac$ Differential equation $$ \left\} e^}}\right\} $$ \left\} \theta }\right\} == Parameter estimation == Given numbers $\backslash \_^n$ drawn from a half-normal distribution, the unknown parameter $\backslash sigma$ of that distribution can be estimated by the method of maximum likelihood, giving : $\backslash hat\; \backslash sigma\; =\; \backslash sqrt$ 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「Half-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.