On permutation and combination in probability and statistics

One 、 A representation of an arrangement

n Choose from different objects r individual （0≤r≤n） The total number of different permutations of is recorded as ：

Be careful , In the mark here n And r The up-down order of is opposite to the general definition of permutation .

Two 、 Combined representation

n Choose from different objects r individual （0≤r≤n） The total number of different combinations of is recorded as ：

Be careful , In the mark here n And r The up-down order of is also opposite to the general combination definition . Therefore, a more general notation will be used in probability statistics to represent this combination ：

Be careful ：

1. The above formula （2.3） in , The combined number sign on the left is called Combination coefficient ;
2. In probability and Statistics , as long as r Is a nonnegative integer ,n The above formula holds for any real number , for example n=-1 when ：
   

3、 ... and 、 Several formulas related to combination

1. The combination coefficient is also called binomial coefficient , Because it appears in the well-known binomial expansion formula below ：
   

2. Another useful formula
   
   notes ： When we take the left side of the equation j=k when , In the formula on the right, the two formulas multiplied by each other get all x^k And then we can get the formula 2.5.

3. n Different objects are divided into k（0≤k≤n） Pile up , The number of each stack is r1,r2,…,rk,（r1,r2,…,rk Is a nonnegative integer , The sum is n）, The total number of different combinations is ：n!/（r1!..rk!）.
   Be careful ：
   1>、 Here, the order of heaps is different, which can be regarded as different division , Such as a、b、c、d、e Divided into three piles , be (ac)、(b)、(de) and (b)、(ac)、(de) Count as two different divisions ;
   2> This combination formula is also called polynomial coefficient , Because it is (x1+…+xk)ⁿ In the expansion of x1^r1…xk^rk The coefficient of this term .

Four 、 Summary

The reason for introducing the knowledge of permutation Statistics , On the one hand, the permutations and combinations in probability and statistics are different from those learned before , In addition, permutation and combination are the main tools of classical probability .

say concretely , Permutations in probability statistics are slightly different from permutations learned before learning probability statistics , And in the combination calculation n choose r Of n The value range is extended to the whole real number field .

In addition, several permutation and combination calculation formulas introduced above are common tools for classical probability , Skilled application of these formulas is helpful to the rapid calculation of classical probability .

Please refer to the column for more mathematical basis of artificial intelligence 《 Mathematical fundamentals of artificial intelligence 》.

It's not easy to write , Please support ：

If you can get something from reading this article , Please give me some compliments 、 Comment on 、 Collection , Thank you for your support ！

Pay column on the old ape

1. Paid column 《https://blog.csdn.net/laoyuanpython/category_9607725.html Use PyQt Developing graphical interfaces Python application 》 Special introduction based on Python Of PyQt Basic course of graphical interface development , The corresponding article directory is 《 https://blog.csdn.net/LaoYuanPython/article/details/107580932 Use PyQt Developing graphical interfaces Python Application column Directory 》;
2. Paid column 《https://blog.csdn.net/laoyuanpython/category_10232926.html moviepy Audio and video development column ) Detailed introduction moviepy The class correlation method of audio and video clip synthesis processing and the use of correlation method to process related clip synthesis scene , The corresponding article directory is 《https://blog.csdn.net/LaoYuanPython/article/details/107574583 moviepy Audio and video development column Directory 》;
3. Paid column 《https://blog.csdn.net/laoyuanpython/category_10581071.html OpenCV-Python Difficult problem set for beginners 》 by 《https://blog.csdn.net/laoyuanpython/category_9979286.html OpenCV-Python Graphics and image processing 》 The accompanying column of , It is the author's opinion of OpenCV-Python Some problems encountered in the study of graphics and image processing , The relevant data are basically the results of repeated studies of old apes , be conducive to OpenCV-Python Beginners have a deeper understanding of OpenCV, The corresponding article directory is 《https://blog.csdn.net/LaoYuanPython/article/details/109713407 OpenCV-Python Catalog of difficult problem set for beginners 》
4. Paid column 《https://blog.csdn.net/laoyuanpython/category_10762553.html Python Introduction to reptiles 》 From the perspective of Xiaobai, an Internet front-end developer, this paper introduces the knowledge and skills of crawler development , Including the basics of getting started with crawlers , And crawling CSDN The article information 、 Blogger information 、 Like the article 、 Comments, etc .

The first two columns are suitable for a certain amount of Python Xiaobai readers who are basic but have no relevant knowledge learn , In the third column, please combine 《https://blog.csdn.net/laoyuanpython/category_9979286.html OpenCV-Python Graphics and image processing 》 Learning to use .

For lack of Python Basic colleagues , Through the free column of the old ape 《https://blog.csdn.net/laoyuanpython/category_9831699.html special column ：Python Basic course catalog ） Learn from scratch Python.

If you are interested and willing to support the old ape readers , Welcome to the paid column .

Old ape Python, Learn from the old ape Python！

* ░ Go to the old ape Python Blog Directory https://blog.csdn.net/LaoYuanPython ░