Suppose the distribution law of discrete random variables is: if there is absolute convergence, there is.
Suppose the density function of continuous random variables is:. If the integral is absolutely convergent, it is the mathematical expectation of random variables. Remember as.
Suppose the density function of continuous random variables is:. If the integral is absolutely convergent, then:
The way to prove it: you can't bring the brain into the formula.
Similar to the definition of conditional distribution, conditional expectation is its expectation under some additional conditions, which can be written as, if there is only one random variable, it can be written as.
If the joint probability density of random variables is known, it can be defined as: given first, the conditional density function of is defined by expectation:
Conditional expectation reflects the average value that changes with the change of value. Statistically, the function of conditional expectation is usually called the "regression function" of pairs.
Combined with the meaning of the total probability formula, we can know that the expectation of a variable should be equal to the weighted average of its conditional expectation pairs, namely:
The proof of formula (1.5) is as follows:
In the formula (1.6), the value of can be written as:
To sum up, the formula (1.5) is proved.
Suppose it is a random variable, and if it exists, it is the expectation of the function of the random variable in popular terms.
Then bring into the formulas (1. 1) and (1.2), we can see that the variance is
Expand:
Covariance:
Correlation coefficient:
Definition of moment: Let it be a random variable, a constant and a positive integer, then this quantity is called the moment of order about. Then, when, it is called "origin moment", and when, it is called central moment.
Assuming that it obeys the standard normal distribution, it is called statistics.
Obey the distribution with degrees of freedom, and record it as.
Let it obey the standard normal distribution, and what obeys it is called statistics.
Obey the distribution with degrees of freedom, and record it as.
Setting obedience, obedience is called statistics.
Obey the distribution with degrees of freedom, and record it as.