Mean Variance and standard deviation"

When I was reading foreign textbooks, I found a concept of Mean Squared Error (mean variance). I went to Zhejiang University's book probability theory and mathematical statistics, the mean variance and standard deviation are defined as a concept above, but the definitions in the foreign language textbooks are quite different. After reading this article, this concept continues to emerge. Therefore, Baidu encyclopedia found that the standard deviation and mean variance were listed as a concept. After querying Wikipedia, we found that the standard deviation and mean variance are not a concept. It is inferred that the expression of mean variance in the textbooks of probability theory of Baidu encyclopedia and Zhejiang University is incorrect.

Mean Squared Error ):

If is a vector of N predictions, and
Is the vector of the true values, then the MSE of the predictor is:

This is a known, computed quantity given a participant sample (and hence is sample-dependent ).

The MSE of
Estimator with respect to the unknown parameter
Is defined

This definition depends on the unknown parameter, and the MSE in this sense is a property of an estimator (of a method of obtaining an estimate ).

The mse is equal to the sum of
Variance and the squared
Bias of the Estimator or of the predictions. In the case of the MSE of an estimator,^[2]

The MSE thus assesses the quality of an estimator or set of predictions in terms of its variation and degree of bias.

Since mse is an expectation, it is not a random variable. It may be a function of the unknown parameter, but it does not depend on any random quantities.
However, when mse is computed for a particle estimator of the true value of which is not known, it will be subject to estimation error. In a Bayesian
Sense, this means that there are cases in which it may be treated as a random variable.

Standard deviation (standard deviation ):

LetXBe
Random Variable with mean valueμ:

Here the operatorEDenotes the average or
Expected ValueX. ThenStandard DeviationOfXIs the quantity

That is, the standard deviationσ(SIGMA) is the square root of thevariance
X, I. e. It is the square root of the average value (X−μ) 2.

The standard deviation of A (univariate) probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation,
Since these expected values need not exist. For example, the standard deviation of a random variable that follows acauchy distribution is undefined because its expected
ValueμIs undefined.

For details, see Wikipedia:

Http://en.wikipedia.org/wiki/Mean_squared_error

Http://en.wikipedia.org/wiki/Standard_Deviation