數學問題~

圖片參考：http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Normal_distribution_pdf.png/325px-Normal_distribution_pdf.png

The green line is the standard normal distribution
綠線代表標準常態分佈
The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. It is a family of distributions of the same general form, differing in their location and scale parameters: the mean ("average") and standard deviation ("variability"), respectively. The standard normal distribution is the normal distribution with a mean of zero and a variance of one (the green curves in the plots to the right). It is often called the bell curve because the graph of its probability density resembles a bell.
正態分佈（normal distribution），又名高斯分佈（Gaussian distribution），是一個在各領域，數學、物理及工程上，皆非常重要的概率分佈，在統計學的許多方面有著重大的影響力。若隨機變量X服從一個數學期望為μ、標準方差為σ2的高斯分佈，記為：


圖片參考：http://upload.wikimedia.org/math/f/7/8/f7898dd28cc9e34d4e932e3636da1647.png

則其概率密度函數為


圖片參考：http://upload.wikimedia.org/math/7/9/a/79af499be9466b7dce2cf8ac19fa0a07.png

正態分佈的期望值μ決定了其位置，其標準差σ決定了分佈的幅度。因其曲線呈鐘形，因此人們又經常稱之為鐘形曲線。我們通常所說的標準正態分佈是μ = 0,σ = 1的正態分佈(見右圖中綠色曲線)。
The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem (the proof of which requires advanced undergraduate mathematics). Many psychological measurements and physical phenomena (like photon counts and noise) can be approximated well by the normal distribution. While the mechanisms underlying these phenomena are often unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation.
The normal distribution also arises in many areas of statistics. For example, the sampling distribution of the sample mean is approximately normal, even if the distribution of the population from which the sample is taken is not normal. In addition, the normal distribution maximizes information entropy among all distributions with known mean and variance, which makes it the natural choice of underlying distribution for data summarized in terms of sample mean and variance. The normal distribution is the most widely used family of distributions in statistics and many statistical tests are based on the assumption of normality. In probability theory, normal distributions arise as the limiting distributions of several continuous and discrete families of distributions.
正態分佈是自然科學與行為科學中的定量現象的一個方便模型。各種各樣的心理學測試分數和物理現象比如光子 計數都被發現近似地服從正態分佈。儘管這些現象的根本原因經常是未知的， 理論上可以證明如果把許多小作用加起來看做一個變量,那麼這個變量服從正態分佈(在R.N.Bracewell的Fourier transform and its application中可以找到一種簡單的證明)。正態分佈出現在許多區域 統計: 例如, 採樣分佈均值是近似地正態的, 既使被採樣的樣本總體並不服從正態分佈。另外, 正態分佈信息熵 在所有的已知均值及方差的分佈中最大，這使得它作為一種均值以及方差已知的分佈的自然選擇。正態分佈是在統計以及許多統計測試中最廣泛應用的一類分佈。在概率論, 正態分佈是幾種連續以及離散分佈的極限分佈。
Details please refer to the following English and Chinese links:
http://en.wikipedia.org/wiki/Normal_distribution
http://zh.wikipedia.org/wiki/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83
I hope I can help.

常態分佈（normal distribution），又名高斯分佈（Gaussian distribution），是一個在各領域，數學、物理及工程上，皆非常重要的機率分佈，在統計學的許多方面有著重大的影響力。