Re-learn a little understanding of probability theory (continuous update)

This is a little bit of understanding about the probability of re-learning after N years of graduation. It takes a lot of time and tragedy to return to school... Therefore, students should hurry up at school to consolidate their foundations.

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Zhejiang University-probability theory and mathematical statistics version 4

Chapter 1 Basic concepts of Probability Theory

1. random test

2. Sample Space and random events

3. Frequency and Probability

4. equipotential probability (classical summary)

5. Conditional Probability

6. Independence

Chapter 2 variables and their distribution

1. Random Variables

2. Discrete Random Variables and Their Distribution Laws

• Meaning: The value is limited, or the number of columns can be unlimited.
• Bernuoli lab

Condition: 1. There are only two results

2. The results of each experiment do not affect each other.

Typical Example: coin throwing

• Binary distribution (0-1 distribution), non-A is B's model. The name comes from the two extensions.

Study content: Relationship between N discrete events and Probability

Typical Example: shooting with a hit rate of 0.02 and independent shooting with a probability of hitting at least two times. Result 0.9972.

Practical Significance: 1. Although the probability of 0.02 is very small, 400 N-weight bernouli experiments were conducted to make the incidence basically an inevitable event;

2. If the final result does not occur (small probability event, <0.003), the hit rate (probability) is less than 0.02;

• Poisson distribution

Study content: probability of event occurrence within a period of time

With two distributions: 1. Divide a period of time into infinite vertices. The probability of event occurrence at each point is the two distributions.

2. the abstract mathematical model is that when n is large enough, Poisson can simulate the two-term distribution.

Poisson theorem: it is easy to understand how to use a student to go to the dining room. Here, we can understand that PN is the probability of each student going to the dining room, and N is the total number of students, lambda is the number of students who finally eat in the canteen.

3. Random Variable Distribution Function

• Distribution function f (x): Describes the statistical regularity of random variables and can be understood using the timeline.

4. Continuous Random Variables and their probability density

• Probability Density: it can be understood as the line density in physics. The area under the probability density curve is 1.

 

• Even Distribution: only related to the Interval Length.

 

• Exponential Distribution:

Non-memory: how long the light bulb has been used, regardless of the total time it can be used, that is, the light bulb has no memory for how long it has been used.

Application: 1. (electronic) Product Life, generally in line with the exponential distribution (under high reliability)

2. It can be used to indicate the interval at which independent random events occur.

• Normal Distribution:

This is too common.

5. Distribution of Random Variable Functions

Chapter 3 Multidimensional random variables and their distribution

1. Two-Dimensional Random Variables

2. Edge Distribution

• In this case, XY has a condition of + ∞ and another distribution probability function, which is reflected in the bottom and rightmost aspects of the data table.

3. conditional distribution

4. Independent Random Variables

5. Distribution of functions of two Random Variables

Chapter 4 digital features of Random Variables

1. mathematical expectation

• Remove the two words of mathematics. It is easy to understand. It is a combination of the expected values of probability. It is usually better because it is often used in the areas of income, gambling, and waiting for a car and a horse.

2. Variance

3. covariance and Correlation Coefficient

4. Moment and covariance matrix