Understand expectations (mean / estimate) and variances

Preface

The formula will be given at the beginning , Then combine the expectation and variance of normal distribution , To understand them more specifically .

expect

First , expect full name “ Mathematical expectation ”, also called mean value or Estimated value , Pay attention to and Average It's not a concept . I'll talk about it later .

Speaking of expectations , We have to talk about the distribution column first .

X It represents the corresponding value of different kinds of events ,P Indicates the probability of finding events per hour .

What is the value corresponding to the event , This is actually our own definition , For example, throw the front 1 branch .

Another example ：

  The score distribution here should be ：

  Then there is the distribution column , You can expect , Or to find the mean / Estimated value . Here we use EX Expressing expectations ：

  Where is the distribution column , It's very simple to sum expectations .( Pictured above ： Multiply and add ). Here we also see the difference between it and the average , The average value is the statistics of what has happened , Averaging . Expectation is the distribution of known probability , Estimate the most likely value .

That's for , Further understand expectations , Introduce here , God's rule —— Normal distribution ：

  Here we can see , expect EX and X The shaft is hooked , In the normal distribution , When X Equal to expectation , At this time, the corresponding Y Value is the largest , But pay attention here , In the normal distribution ,Y The value is not X Corresponding probability value , Normal distribution is not a curve of probability , But the density curve of probability , Here's my understanding （ Not necessarily ）： hypothesis X It can be infinitely subdivided , This curve is a positive distribution of height , Suppose a person's height is 1.60025499 Infinite subdivision , The height of the other person is 2.045648842 Infinite subdivision . Then these two heights , Whose probability is high ？ They are infinitely close 0. But if it's a range , For example, the height is 1.65 About meters and height in 2.2 About meters , Whose probability is high , That is obviously the former .

Here, the area below the curve is the probability P. The total area under the curve is 1.

This graph corresponds to the probability density function f(X), therefore Y The value corresponds to the cumulative distribution function F(X) First derivative of , namely F(X) Instantaneous rate of change . The actual meaning is that X Corresponding density value . That is, the place where the probability accumulates fastest .

So here , We found that Y Value is actually a very abstract , That's what I know X But I got one Y, But this Y It doesn't mean much to me , I know at this time Y Maximum . So we often look the other way ！ We know the biggest Y, Then look at the corresponding X How much is the , and X The meaning of is specific . such as Y Maximum time X=1.6, that 1.6 Expectation is the mean . When the sample is determined , The mean value and the mean value are equal .

Class reference ：

​​​​​​ Is the mean and expectation the same thing ？ - You know (zhihu.com)https://zhuanlan.zhihu.com/p/311896697 Speaking of this , The part about expectation is over .

When is the variance ？

First, the formula is given

  From the formula , Variance is related to expectation and probability ,（ Expectation itself is also related to probability ）. Variance describes a set of data （ sample ） The degree of dispersion of .

  Here we can combine the deviation , To compare and understand the variance .

Then by simplifying the formula , We finally get the formula , among D(x) Variance ：

  The simplification process is as follows ：

  Here I also introduce normal distribution , Further understand variance .

When Sigma is determined , There is another one 3 Sigma principles , You can see ：（ There is also a concept of standard deviation , This is sigma , The square of sigma is variance ）

   It can be seen here that when the variance is larger , In the same area , Smaller area , That is, the smaller the probability , This shows that the more scattered the data .

Summary ： 

        1 That's in deep learning , This large variance model may need more classification to better define .

        2 Variance and expectation are strongly correlated with probability , The corresponding value of each event needs to be multiplied by the probability P, Doing other calculations .

        

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Reference material ：

​​​​​​【#11】【28 Days beat math in the college entrance examination 】 Distribution column Mathematical expectation variance | High school math | College entrance examination mathematics _ Bili, Bili _bilibili

【 probability · Normal distribution 】 What is this ？10 Minutes to get started completely ！_ Bili, Bili _bilibili

Learn from Li Mu AI Personal space _ Bili, Bili _bilibili