不知道這個東西是不是只會用這一次，反正搞清楚了，就留下來吧。
參考文獻：https://en.wikipedia.org/wiki/Log-normal_distribution
https://blog.csdn.net/Eric2016_Lv/article/details/53286434
在概率論中，對數正態分布是一種連續概率分布，其隨機變量的對數服從正態分布。
對數正態分布圖
wiki原話：
A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of ln(X) are specified.
The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).
If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal.
我的理解是：從統計學角度理解對數正態分布是這樣的，在自然界有很多事物有增長速度很慢，甚至可以忽略不計（small percentage changes），但是其效果是對整個事物的影響，即每次增長都是對前面增長的乘積運算，但如果我們把他放入對數域，則可以放大他們的增長效果。
假設：x1,x2,...,xk表示第i個單位時間的單位增長率，則x1,x2,...xk大於等於0，令zi=log(xi)表示xi的對數
顯然有：
因為x1,x2,...xk獨立同分布，顯然z1,z2...zk也是獨立同分布，則根據中心極限定理（當樣本量足夠大時，樣本均值的分布（變量和的分布）慢慢變成正態分布）有：
這也就符合上面wiki第一段的意思了吧。其中一個典型的例子是股票投資的長期收益率，它可以看作是每天收益率的乘積（雖然他不是自然界的）。
對數正態分布的概率密度函數為:
期望值和方差分別為：
其中，μ和σ分別是變量對數的平均值和標准差。 而對於參數μ和σ可以用極大使然估計來求解：
