Summary of probability theory and mathematical statistics (1)

So far, Mr. Chen has read the most cordial book on probability theory and mathematical statistics, which is nothing more than that of Mr. Chen. Mr. Chen has made a lot of originally complex content so clear in a concise tone, in addition, it is not based on this knowledge, but can be introduced together with the knowledge system before and after.

This book is titled probability theory and mathematical statistics. It mainly talks about two major knowledge systems. The first half (the first three chapters) is about probability theory, and the second half (the last three chapters) is about mathematical statistics.

From the knowledge point of view, Chapter 1 describes the probability of an event, including what is probability (what is probability), classical probability calculation, event calculation, conditional probability, and probability independence. In this chapter, events are the basis of the entire probability. How to define probability is also the key content of the evolution of the entire probability theory knowledge system, including classical probability (geometric probability) statistical definitions of probabilities and definitions of probabilities. The basis of classical probability is equality possibility. The geometric probability is extended to an infinite number based on classical probability. The statistical definition considers non-equality possibility; the theory of probability is based on the theory of set. The theory of set to describe probability also makes the probability into a strict mathematical category.

Section 1.2 describes the calculation of Classical probability. Classical probability calculation is a common calculation method. It is based on Permutation and combination. It also tests the knowledge of a person in consideration of problems and permutation and combination calculation, it is important to consider the overall computing and event computing.

Section 1.3 describes event operations, conditional probabilities, and independence. A complex event is divided into simple events (or atomic events), or the probability of a complex event is calculated based on the event calculation method. Any operation is composed of three parts: Operation object, operator, and operation result. The calculation object of an event is an event, and there are many operators, including the relationship between events.

1.3.1 (contains, and equal), meaning a occurs => B occurs, that is, B contains a. If a \ subseteq B and B \ subseteq A, it indicates a = B

1.3.2 events are mutually exclusive and opposite. Mutual Exclusion is an independent event. An important event of mutual exclusion is an opposite Event B = {A does not happen}, and B is a opposite event.

1.3.3 sum of events: the sum of events, either of which occurs at will,

1.3.4 addition theorem of Probability: the probability of the sum of multiple mutex events is equal to the sum of probabilities of each event.

1.3.5 event product and event difference: the event product is the intersection of events, and the event difference

1.3.6 conditional probability: What is the probability of occurrence of event? P (B | A) = P (BA)/P (A) and P (A/B) = P (AB)/P (B ), P (B | A) P (A) = P (A | B) P (B) is introduced, which is one of the basis of Bayesian formulas.

1.3.7 independence of events and probability multiplication theorem: In the past, many people confused the independence of events and the mutual exclusion of events. The mutual exclusion of events indicates that two events are active and dead. A occurs, B cannot occur. independence is the relationship between A and B, that is, P (A | B) = P (A), regardless of whether B occurs, the probability of occurrence of a is the same. Another formula is available: P (AB) = P (a) P (B ). Spread to multiple events: P (A1, A2,... an) = P (A1) P (A2)... P (). The concept of independence is very important.

1.3.8 full probability formula and Bayesian formula: The full probability formula considers the probability of a complete subevent group. P (A) = P (Ab1) + P (Ab2) +... + P (ABN); although the Bayesian formula is only the derivation of the full probability formula and the definition of the conditional probability, it is of great significance in reality. The full probability formula is derived from the cause, the Bayesian formula is based on the results. For example, the naive Bayesian Classification in the machine learning field and the spam classification example are as follows, calculate the probability that the keyword appears as a spam email and the probability that the keyword appears as a non-spam email to classify the email. This is an application of Bayesian formulas. Of course, Bayesian statistics is much more complex than the previous example. It has become a school that considers the anterior probability, in addition to the overall information and sample information, prior information must be considered. Although simple, it is very important.

Summary of probability theory and mathematical statistics (1)