Log-normal distribution

I don't know if this item will only be used this time. I figured it out, just stay.

References: https://en.wikipedia.org/wiki/Log-normal_distribution

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In probability theory, the logarithm normal distribution is a continuous probability distribution, and the logarithm of the random variable follows a normal distribution.

　　　　　　　　　　　　　　

Log Normal Distribution chart

Original wiki statement:

A log-normal process is the statistical realization of the multiplicative product of specified 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 variateXFor 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 extends natural growth processes are driven by the accumulation of variable small percentage changes. these become Additive on a log scale. if the effect of any one change is negligible,Central limit theoremSays 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 too 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.

My understanding is: from a statistical point of view, the log normal distribution is like this. In nature, there are many things that are growing slowly, or even negligible (small percentage changes),The effect is the impact on the entire thing, that is, every growth is the product operation of the previous growth,But if we put him in the logarithm field, we can enlarge their growth effect.

Suppose: x1, x2 ,..., XK indicates the unit growth rate of the I unit time, then x1, x2 ,... XK is greater than or equal to 0, so Zi = Log (xi) indicates the logarithm of Xi

Apparently there are:

Because x1, x2 ,... XK is independently distributed, obviously Z1, Z2... ZK is also independent of the same distribution, so according to the central limit theorem (when the sample size is large enough, the distribution of the sample mean (the distribution of variables and) gradually becomes a normal distribution:

　　　

This is in line with the meaning of the first section of the Wiki above. A typical example is the long-term return rate of stock investment. It can be seen as the product of the daily return rate (although not in nature ).

The probability density function of the logarithm normal distribution is:

 

Expected values and variance are:

Here, μ and σ are the mean and standard deviation of the logarithm of the variable. For the μ and σ parameters, we can use a great deal of estimation to solve the problem:

 

Log-normal distribution