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11.2. Linearize the function of one random argument

Lecture



In practice, the need to linearize the function of one random argument is relatively rare: usually the cumulative effect of several random factors has to be taken into account. However, from methodological considerations it is convenient to start from this simplest case. Let there be a random variable   11.2.  Linearize the function of one random argument and its numerical characteristics are known: expectation   11.2.  Linearize the function of one random argument and variance   11.2.  Linearize the function of one random argument .

Assume that practically possible values ​​of a random variable   11.2.  Linearize the function of one random argument limited to   11.2.  Linearize the function of one random argument i.e.

  11.2.  Linearize the function of one random argument .

There is another random variable.   11.2.  Linearize the function of one random argument related to   11.2.  Linearize the function of one random argument functional dependence:

  11.2.  Linearize the function of one random argument , (11.2.1)

and function   11.2.  Linearize the function of one random argument although not linear, it differs little from linear in the segment   11.2.  Linearize the function of one random argument .

Required to find the numerical characteristics of   11.2.  Linearize the function of one random argument - expected value   11.2.  Linearize the function of one random argument and variance   11.2.  Linearize the function of one random argument .

Consider the curve   11.2.  Linearize the function of one random argument Location on   11.2.  Linearize the function of one random argument (fig. 11.2.1) and replace it with an approximately tangent at the point   11.2.  Linearize the function of one random argument with abscissa   11.2.  Linearize the function of one random argument . The tangent equation is:

  11.2.  Linearize the function of one random argument . (11.2.2)

  11.2.  Linearize the function of one random argument

Fig. 11.2.1

Suppose that the interval of practically possible values ​​of the argument   11.2.  Linearize the function of one random argument so narrow that within this interval the curve and the tangent differ little, so that the curve section can practically be replaced by the tangent section; in short, on the site   11.2.  Linearize the function of one random argument function   11.2.  Linearize the function of one random argument almost linear. Then random variables   11.2.  Linearize the function of one random argument and   11.2.  Linearize the function of one random argument approximately related linear dependence:

  11.2.  Linearize the function of one random argument ,

or denoting   11.2.  Linearize the function of one random argument ,

  11.2.  Linearize the function of one random argument . (11.2.3)

To the linear function (11.2.3), one can apply the well-known methods of determining the numerical characteristics of linear functions (see   11.2.  Linearize the function of one random argument 10.2). The mathematical expectation of this linear function will be found by substituting in its expression (11.2.3) the mathematical expectation of the argument   11.2.  Linearize the function of one random argument equal to zero. We get:

  11.2.  Linearize the function of one random argument . (11.2.4)

Variance of magnitude   11.2.  Linearize the function of one random argument determined by the formula

  11.2.  Linearize the function of one random argument . (11.2.5)

Turning to the standard deviation, we have:

  11.2.  Linearize the function of one random argument . (11.2.6)

Formulas (11.2.4), (11.2.5), (11.2.6), of course, are approximate, since the replacement of a nonlinear function by a linear one is also approximate.

Thus, we solved the task and came to the following conclusions.

To find the expectation of an almost linear function, you need to substitute its expectation instead of an argument in the function expression. To find the variance of an almost linear function, the variance of the argument must be multiplied by the square of the derivative of the function at the point corresponding to the expectation of the argument.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis