MindMap Gallery Probability and Statistics Chapter 5 Numerical Characteristics of Random Variables
Overview Chapter 5: Numerical characteristics of random variables, including mathematical expectation, variance, mathematical expectation and variance, covariance and correlation coefficient of commonly used random variables, etc.
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This Valentine's Day brand marketing handbook provides businesses with five practical models, covering everything from creating offline experiences to driving online engagement. Whether you're a shopping mall, restaurant, or online brand, you'll find a suitable strategy: each model includes clear objectives and industry-specific guidelines, helping brands transform traffic into real sales and lasting emotional connections during this romantic season.
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The ice hockey schedule for the Milano Cortina 2026 Winter Olympics, featuring preliminary rounds, quarterfinals, and medal matches for both men's and women's tournaments from February 5–22. All game times are listed in Eastern Standard Time (EST).
General Chapter 5 Numerical Characteristics of Random Variables
5.1 Mathematical expectations
mean of r.v.
When n is large enough, the frequency tends to the probability value, so we use probability to replace the frequency and introduce the concept of mathematical expectation (mean). Mathematical expectation is the generalization of the average.
definition:
The mathematical expectation of discrete random variable X:
The mathematical expectation of the function of discrete random variable X:
Mathematical expectation of continuous random variable X:
The mathematical expectation of the function Y=g(X) of the continuous random variable X:
Mathematical expectation of a function of a random vector:
properties of mathematical expectations
Assuming C is a constant, then E(C)=C;
Suppose C is a constant and X is a random variable, then we have
Let X and Y be arbitrary random variables,
Suppose X and Y are mutually independent random variables, then we have
5.2 Variance
r.v. The average deviation of the value from the mean
The concept of variance
definition:
Calculation formula:
properties of variance
Assuming C is a constant, then D(C)=0;
Assuming k is a constant and X is a random variable, then we have
Assume that X and Y are independent random variables, then D(X Y)=D(X) D(Y)
5.3 Mathematical expectation and variance of commonly used random variables
Properties of normal distribution
5.4 Covariance and correlation coefficient
Describe a certain relationship between two r.v.
Covariance
When X and Y are independent, then D(X Y)=DX DY (variance property 3). When X and Y are not independent, then the property 5 above is
Correlation coefficient
definition
nature:
moment, covariance matrix
The concept of moment: Moment is a general or collective name for some numerical characteristics. In probability theory and mathematical statistics, moments occupy an important position.
covariance matrix
Distribution function (distribution law, probability density): Completely describes all probability laws of r.v. However, in practical applications, it is difficult or unnecessary to understand all its probability laws. Just know some statistical properties of r.v.