Talk about probability theory

I. Introduction

Recently I started to learn about random processes. In fact, it is not a rigorous random process. In fact, I am learning about random signal processing. I feel that random events can still describe the natural world and it is a good tool. My major is not in communication, and I may not need to study very deeply. As the basis of random processes-probability theory and digital signal processing feel that the foundation is not very solid. Therefore, I want to sort out my rough understanding of probability theory. I have a system of probability theory in my mind to facilitate individual understanding and application.

First, when talking about probability theory, we must first know why such a thing exists and what is its application field? Next, since it is probability theory, what is probability? Whose probability is it? How do we describe this probability or how to get this probability? In order to answer these questions, the following is divided into several points to expand:

Probability space
Probability (classical probability, conditional probability-total probability formula and Bayes formula & independent events)
Numerical characteristics of random variables
Distribution function of random variable (distribution function of random variable & function distribution of random variable & characteristic function of random variable)
Limit theorem

Second, the probability space

Many events in the world are deterministic, which we call deterministic phenomena, such as the sun rising from the east; but there are also many phenomena that are uncertain before they occur, and we don’t know the results of their occurrence. For example, will it rain tomorrow? How many will the dice roll? The existence of probability theory was born to describe such uncertain phenomena. Why describe such uncertain phenomena? Within the scope of my understanding, it is because humans want to understand some phenomena that were originally considered uncontrollable and unknown, and to grasp the laws of their development, so as to be able to predict and control them. Therefore, in order to understand an unknown phenomenon, we can imagine putting this phenomenon in a black box, and then slowly study the black box, that is, slowly study and grasp this unknown phenomenon. This black box is what we often call the probability space.

The probability space is composed of three elements-S, F, P. Let's explain what S, F, and P are with specific examples. First of all, we want to understand what will appear in the sky tomorrow? This is the question we asked about the black box, that is, the probability space. Then we will draw a note from the black box and see what the answer is written on it. We call the process of drawing a note as an experiment. The words that appear on all the pieces of paper in the black box, such as clouds, such as the sun (all possible results in the experiment), are called random events. All the results of random events form S-sample space . Here we can notice that random events have three characteristics:

Test can be the same conditions was repeated for
We can know in advance the results of all possible random events ;
But we don’t know which random event will happen.

But sometimes the answer on the note may be more invalid and amazing, for example, it says that there will be soft sister coins in the sky tomorrow. You will sigh: Oh! This is really a good thing! But in fact, we are very clear that there will be no soft sister coins in the sky. Is it true that the black box is full of these nonsense notes?

No, the note may also say that there will be a big sun in the sky tomorrow, remember to take a parasol. If you didn't believe it, you might be miserable the next day, and you will be tanned for several degrees. It may also be written on the note, my dear, tomorrow morning the weather will be very good, the sky is the big sun, but this note and another note accidentally said that there are thick black clouds in the sky in the afternoon may rain heavy rain Together! As a result, the next day you found out, eh, these two "accidentally" sticky notes were right again! In this way, the black box is really good. For tomorrow, the sky may be a big sun, there may be stars, there may be thick black clouds so that it will rain. These answers may be given on these papers. These answers are acceptable in our understanding. Within the scope, we also think they may indeed appear. This is F-a collection of random events . F may be a single element (for example, a note telling you that there will be a big sun tomorrow); it may also be multiple elements, two pieces of paper stuck together (the big sun in the morning and the rain in the afternoon). F also has three characteristics:

The collection must contain impossible events;
If events E{1}, E{2}, E{3}, E{4}∈F, then E{1,2,3,4}∈F;
If the event E ∈ F, then the complement of E ∈ F.

At this point, are you confused, eh, why do you define S and F separately? Are they not all possible results? In my opinion, extracting F separately to define it actually has two effects: 1) Make the sample space descriptive . Why is this? Why does the sample space need F to describe it? Recall the example of the soft sister coin just mentioned. In fact, there are many examples of such nonsensical ones. If we list them one by one, we may have been doing this for a lifetime. And F allows us to draw only a few of them. For example, I draw soft sister coins, which makes our description of random events countable and descriptive; 2) can make full use of all possible random events. F is defined as a set, that is, it can contain one random event or two random events. For example, we gave the example of the morning sun and rain in the afternoon. The description of the set makes our description of the probability space more realistic.

Finally, we will introduce the third element P of the probability space, which is developed around F, which is the probability of F occurrence. P also has three characteristics:

0 ≤ P(E))≤ 1；
P (Ω) = 1 ；
If E{1}, E{2}, E{3}...orthogonal, P(E{1} + E{2} + E{3} + ......) = ∑P(E{n}).

References in this chapter:

[https://www.zhihu.com/question/20642770/answer/67964978]:

The second part strives for more next week! !