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12.3. The distribution law of a nonmonotonic function of one random argument

Lecture



There is a continuous random variable   12.3.  The distribution law of a nonmonotonic function of one random argument with distribution density   12.3.  The distribution law of a nonmonotonic function of one random argument ; other value   12.3.  The distribution law of a nonmonotonic function of one random argument associated with   12.3.  The distribution law of a nonmonotonic function of one random argument functional dependence:

  12.3.  The distribution law of a nonmonotonic function of one random argument ,

and function   12.3.  The distribution law of a nonmonotonic function of one random argument Location on   12.3.  The distribution law of a nonmonotonic function of one random argument possible values ​​of the argument are not monotonous (Fig. 12.3.1).

  12.3.  The distribution law of a nonmonotonic function of one random argument

Fig. 12.3.1.

Find the distribution function   12.3.  The distribution law of a nonmonotonic function of one random argument magnitudes   12.3.  The distribution law of a nonmonotonic function of one random argument . To do this, again, hold the line   12.3.  The distribution law of a nonmonotonic function of one random argument parallel to the abscissa at a distance   12.3.  The distribution law of a nonmonotonic function of one random argument from it and select those parts of the curve   12.3.  The distribution law of a nonmonotonic function of one random argument on which the condition is met   12.3.  The distribution law of a nonmonotonic function of one random argument . Let these sections correspond to the sections of the x-axis:   12.3.  The distribution law of a nonmonotonic function of one random argument

Event   12.3.  The distribution law of a nonmonotonic function of one random argument tantamount to hitting a random variable   12.3.  The distribution law of a nonmonotonic function of one random argument on one of the sites   12.3.  The distribution law of a nonmonotonic function of one random argument - all the same on which one. therefore

  12.3.  The distribution law of a nonmonotonic function of one random argument

  12.3.  The distribution law of a nonmonotonic function of one random argument .

Thus, for the distribution function of   12.3.  The distribution law of a nonmonotonic function of one random argument we have the formula:

  12.3.  The distribution law of a nonmonotonic function of one random argument . (12.3.1)

Spacing boundaries   12.3.  The distribution law of a nonmonotonic function of one random argument depends on   12.3.  The distribution law of a nonmonotonic function of one random argument and given the specific form of the function   12.3.  The distribution law of a nonmonotonic function of one random argument can be expressed as explicit functions   12.3.  The distribution law of a nonmonotonic function of one random argument . Differentiating   12.3.  The distribution law of a nonmonotonic function of one random argument by size   12.3.  The distribution law of a nonmonotonic function of one random argument entering the limits of integrals, we obtain the distribution density of   12.3.  The distribution law of a nonmonotonic function of one random argument :

  12.3.  The distribution law of a nonmonotonic function of one random argument . (12.3.2)

Example. Magnitude   12.3.  The distribution law of a nonmonotonic function of one random argument subject to the law of uniform density in the area from   12.3.  The distribution law of a nonmonotonic function of one random argument before   12.3.  The distribution law of a nonmonotonic function of one random argument :

  12.3.  The distribution law of a nonmonotonic function of one random argument

Find the law of distribution of magnitude   12.3.  The distribution law of a nonmonotonic function of one random argument .

Decision. Build a graph of the function   12.3.  The distribution law of a nonmonotonic function of one random argument (fig. 12.3.2). Obviously   12.3.  The distribution law of a nonmonotonic function of one random argument ,   12.3.  The distribution law of a nonmonotonic function of one random argument and in the interval   12.3.  The distribution law of a nonmonotonic function of one random argument function   12.3.  The distribution law of a nonmonotonic function of one random argument not monotonous.

  12.3.  The distribution law of a nonmonotonic function of one random argument

Fig. 12.3.2.

Applying the formula (12.3.1), we have:

  12.3.  The distribution law of a nonmonotonic function of one random argument .

Express limits   12.3.  The distribution law of a nonmonotonic function of one random argument and   12.3.  The distribution law of a nonmonotonic function of one random argument through   12.3.  The distribution law of a nonmonotonic function of one random argument :

  12.3.  The distribution law of a nonmonotonic function of one random argument ;   12.3.  The distribution law of a nonmonotonic function of one random argument .

From here

  12.3.  The distribution law of a nonmonotonic function of one random argument . (12.3.3)

To find the density   12.3.  The distribution law of a nonmonotonic function of one random argument We will not calculate the integrals in the formula (12.3.3), but directly differentiate this expression with respect to the variable   12.3.  The distribution law of a nonmonotonic function of one random argument within the integrals:

  12.3.  The distribution law of a nonmonotonic function of one random argument .

Bearing in mind that   12.3.  The distribution law of a nonmonotonic function of one random argument , we get:

  12.3.  The distribution law of a nonmonotonic function of one random argument . (12.3.4)

Pointing for   12.3.  The distribution law of a nonmonotonic function of one random argument distribution law (12.3.4), it should be stipulated that it is valid only in the range from 0 to 1, i.e. within the limits in which it changes   12.3.  The distribution law of a nonmonotonic function of one random argument with argument   12.3.  The distribution law of a nonmonotonic function of one random argument between   12.3.  The distribution law of a nonmonotonic function of one random argument and   12.3.  The distribution law of a nonmonotonic function of one random argument . Beyond these limits the density   12.3.  The distribution law of a nonmonotonic function of one random argument equals zero.

Function graph   12.3.  The distribution law of a nonmonotonic function of one random argument given in fig. 12.3.3. With   12.3.  The distribution law of a nonmonotonic function of one random argument curve   12.3.  The distribution law of a nonmonotonic function of one random argument has a branch going to infinity.

  12.3.  The distribution law of a nonmonotonic function of one random argument

Fig. 12.3.3.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis