每日文章（七十二） 用常識來理解正態分佈（中英文）

Daily article 72: Understand Normal distribution by common sense

Unlike discrete random variables, continuous random variables have infinite outcomes, so the probability of any specific outcome is 0. Therefore, we don’t care about the probability of a specific outcome for continuous random variables. Rather, we are curious about the probability that the outcome falls within a specific interval, which contains infinite outcomes as well. A frequency distribution demonstrates how the outcomes are distributed.

Normal distribution is also called Gaussian distribution to commemorate the contribution made by the great mathematician, Gaussian. Remember the famous Gaussian Algorithm from middle school? That’s him. Normal distribution is very important in statistics because many statistical results follow the normal distribution. Normal distribution is a symmetric distribution that can be described solely by two parameters: mean μ and standard deviation σ (middle school math again). Mean determines the normal distribution’s middle line and standard deviation determines the level of dispersion. Let’s take a look at the graph below:

The normal distribution is a bell-shape distribution, which means the probabilities of intervals near to mean are greater than that of intervals far away from mean. The result coordinates with our common sense. For example, there are more people of average height and few people are extremely tall or short. We can find random variables of normal distribution in many areas in our daily life，among which the equity return is the most common one in financial world. It’s also pretty straightforward: we have higher chances to get a return around expected return, and lower chances to get extremely large gains or losses.

每日文章（七十二） 用常識來理解正態分佈

與離散隨機變量不同，連續隨機變量有無數種結果，所以任一特定結果的可能性，無限接近於0。因此，我們並不關心連續隨機變量的某一結果的概率，我們希望瞭解的是結果落入某一區間的概率（這一區間也包含無數個可能結果）。頻率分佈描述了這些結果是怎樣分佈的。

正態分佈也叫做高斯分佈，目的是紀念偉大的數學家高斯在研究正態分佈中做出的貢獻。還記得我們在初中學習的高斯算法（首項加末項乘以項數除以2）嗎？就是這個高斯。在統計學中，正態分佈非常重要，很多統計結果都是正態分佈的。正態分佈圖是一個軸對稱的分佈圖，它由兩個參數確定：均值和標準差（仍然是初中數學知識）。均值確定了正態分佈的中線在哪，標準差確定了正態分佈的離散程度。讓我們看看下面的正態分佈圖：

正態分佈是一個鐘的形狀，這意味著相對於遠離均值的區間，該連續變量的結果有更大概率落入接近均值的區間，這其實和我們的常識是一致的。舉例來說，大部分人的身高接近於平均身高，只有少數人特別高或者特別矮。事實上，符合正態分佈的變量在生活中廣泛存在，其中金融領域最常見的就是股票的回報率。這也很容易理解：我們有更高的概率取得一個接近期望收益率的收益率，而大賺或者大虧的概率則比較低。