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8.8. Numerical characteristics of a system of several random variables

Lecture



The distribution law of a system (given by a distribution function or distribution density) is a complete, exhaustive characteristic of a system of several random variables. However, very often such an exhaustive characteristic cannot be applied. Sometimes the limitations of experimental material make it impossible to build the law of distribution of the system. In other cases, the study of the issue with the help of a relatively cumbersome apparatus of the laws of distribution does not justify itself due to the low demands on the accuracy of the result. Finally, in a number of problems, the approximate type of the distribution law (the normal law) is known in advance and it is only necessary to find its characteristics.

In all such cases, instead of the distribution laws, an incomplete, approximate description of a system of random variables using the minimum number of numerical characteristics is used.

The minimum number of characteristics with which the system can be characterized   8.8.  Numerical characteristics of a system of several random variables random variables   8.8.  Numerical characteristics of a system of several random variables It comes down to this:

one)   8.8.  Numerical characteristics of a system of several random variables mathematical expectations

  8.8.  Numerical characteristics of a system of several random variables ,

characterizing the average values;

2)   8.8.  Numerical characteristics of a system of several random variables dispersions

  8.8.  Numerical characteristics of a system of several random variables ,

characterizing their dispersion;

3)   8.8.  Numerical characteristics of a system of several random variables correlation moments

  8.8.  Numerical characteristics of a system of several random variables ,

Where

  8.8.  Numerical characteristics of a system of several random variables ,

characterizing the pairwise correlation of all quantities included in the system.

Note that the variance of each of the random variables   8.8.  Numerical characteristics of a system of several random variables there is essentially nothing more than a special case of the correlation moment, namely the correlation moment of magnitude   8.8.  Numerical characteristics of a system of several random variables same size   8.8.  Numerical characteristics of a system of several random variables :

  8.8.  Numerical characteristics of a system of several random variables .

All correlation moments and variances are conveniently arranged in the form of a rectangular table (the so-called matrix):

  8.8.  Numerical characteristics of a system of several random variables .

This table is called the correlation matrix of a system of random variables.   8.8.  Numerical characteristics of a system of several random variables .

Obviously, not all members of the correlation matrix are different. From the definition of the correlation moment, it is clear that   8.8.  Numerical characteristics of a system of several random variables that is, the elements of the correlation matrix that are located symmetrically with respect to the main diagonal are equal. In this connection, not all the correlation matrix is ​​often filled, but only half of it, counting from the main diagonal:

  8.8.  Numerical characteristics of a system of several random variables .

Correlation matrix composed of elements   8.8.  Numerical characteristics of a system of several random variables , often abbreviated as symbol   8.8.  Numerical characteristics of a system of several random variables .

On the main diagonal of the correlation matrix are the variances of random variables.   8.8.  Numerical characteristics of a system of several random variables .

In the case when random variables   8.8.  Numerical characteristics of a system of several random variables not correlated, all elements of the correlation matrix, except for the diagonal ones, are equal to zero:

  8.8.  Numerical characteristics of a system of several random variables .

Such a matrix is ​​called diagonal.

For the sake of clarity, judgments about the correlation of random variables, irrespective of their dispersion, are often instead of a correlation matrix.   8.8.  Numerical characteristics of a system of several random variables use the correlation matrix   8.8.  Numerical characteristics of a system of several random variables composed not of correlation moments, but of correlation coefficients:

  8.8.  Numerical characteristics of a system of several random variables ,

Where

  8.8.  Numerical characteristics of a system of several random variables .

All the diagonal elements of this matrix are naturally equal to one. The normalized correlation matrix is:

  8.8.  Numerical characteristics of a system of several random variables .

We introduce the concept of uncorrelated systems of random variables (otherwise, uncorrelated random vectors). Consider two systems of random variables:

  8.8.  Numerical characteristics of a system of several random variables

or two random vectors in   8.8.  Numerical characteristics of a system of several random variables -dimensional space:   8.8.  Numerical characteristics of a system of several random variables with components   8.8.  Numerical characteristics of a system of several random variables and   8.8.  Numerical characteristics of a system of several random variables with components   8.8.  Numerical characteristics of a system of several random variables . Random vectors   8.8.  Numerical characteristics of a system of several random variables and   8.8.  Numerical characteristics of a system of several random variables are called uncorrelated if each of the components of the vector   8.8.  Numerical characteristics of a system of several random variables not correlated with each of the components of the vector   8.8.  Numerical characteristics of a system of several random variables :

  8.8.  Numerical characteristics of a system of several random variables .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis