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12.6. Composition of normal laws

Lecture



Consider two independent random variables.   12.6.  Composition of normal laws and   12.6.  Composition of normal laws , subject to normal laws:

  12.6.  Composition of normal laws , (12.6.1)

  12.6.  Composition of normal laws . (12.6.2)

It is required to produce a composition of these laws, i.e., to find the law of distribution of magnitude:

  12.6.  Composition of normal laws .

We apply the general formula (12.5.3) to the composition of the laws of distribution:

  12.6.  Composition of normal laws

  12.6.  Composition of normal laws . (12.6.3)

If we expand the brackets in the exponent of the integrand function and give similar terms, we get:

  12.6.  Composition of normal laws ,

Where

  12.6.  Composition of normal laws ;

  12.6.  Composition of normal laws ;

  12.6.  Composition of normal laws .

Substituting these expressions into the formula we have already encountered (9.1.3):

  12.6.  Composition of normal laws , (12.6.4)

after transformation we get:

  12.6.  Composition of normal laws , (12.6.5)

and this is nothing but a normal law with a center of dispersion

  12.6.  Composition of normal laws (12.6.6)

and standard deviation

  12.6.  Composition of normal laws . (12.6.7)

Moreover, the conclusion can be made much simpler with the help of the following qualitative considerations.

Without opening the brackets and not making the transformations in the integrand function (12.6.3), we immediately conclude that the exponent is a square trinomial with respect to   12.6.  Composition of normal laws kind of

  12.6.  Composition of normal laws ,

where is the coefficient   12.6.  Composition of normal laws magnitude   12.6.  Composition of normal laws not included at all   12.6.  Composition of normal laws included in the first degree, and in the coefficient   12.6.  Composition of normal laws - squared. With this in mind and applying the formula (12.6.4), we conclude that   12.6.  Composition of normal laws there is an exponential function, the exponent of which is a square triple relative to   12.6.  Composition of normal laws and the distribution density of this type corresponds to the normal law. Thus, we arrive at a purely qualitative conclusion: the law of distribution of   12.6.  Composition of normal laws should be normal.

To find the parameters of this law -   12.6.  Composition of normal laws and   12.6.  Composition of normal laws - we use the theorem of addition of mathematical expectations and the theorem of addition of variances. By the theorem of addition of mathematical expectations

  12.6.  Composition of normal laws . (12.6.8)

According to the theorem of addition of variances

  12.6.  Composition of normal laws

or

  12.6.  Composition of normal laws , (12.6.9)

where follows the formula (12.6.7).

Moving from standard deviations to proportional probable deviations, we get:

  12.6.  Composition of normal laws . (12.6.10)

Thus, we have arrived at the following rule: with the composition of normal laws, the normal law is obtained again, and the mathematical expectations and variances (or squares of probable deviations) are summed up.

The rule of composition of normal laws can be generalized to the case of an arbitrary number of independent random variables.

If available   12.6.  Composition of normal laws independent random variables:

  12.6.  Composition of normal laws ,

subject to normal laws with dispersion centers

  12.6.  Composition of normal laws

and standard deviations

  12.6.  Composition of normal laws ,

that magnitude

  12.6.  Composition of normal laws

also subject to normal law with parameters

  12.6.  Composition of normal laws , (12.6.11)

  12.6.  Composition of normal laws . (12.6.12)

Instead of formula (12.6.12), the equivalent formula can be applied:

  12.6.  Composition of normal laws . (12.6.13)

If the random variable system   12.6.  Composition of normal laws distributed according to the normal law, but the magnitudes   12.6.  Composition of normal laws dependent, it is easy to prove, just as before, based on the general formula (12.5.1), that the distribution law

  12.6.  Composition of normal laws

there is also a normal law. The scattering centers are still algebraically added, but for standard deviations, the rule becomes more complex:

  12.6.  Composition of normal laws , (12.6.14)

Where   12.6.  Composition of normal laws - coefficient of correlation of quantities   12.6.  Composition of normal laws and   12.6.  Composition of normal laws .

When adding several dependent random variables, subordinate in their totality to the normal law, the law of the distribution of the sum also turns out to be normal with the parameters

  12.6.  Composition of normal laws , (12.6.15)

  12.6.  Composition of normal laws , (12.6.16)

or in probable deviations

  12.6.  Composition of normal laws , (12.6.17)

Where   12.6.  Composition of normal laws - coefficient of correlation of quantities   12.6.  Composition of normal laws , and summation applies to all different pairwise combinations of quantities   12.6.  Composition of normal laws .

We were convinced of a very important property of a normal law: with the composition of normal laws, a normal law is obtained again. This is the so-called “stability property”. The law of distribution is called stable if the composition of two laws of this type again results in a law of the same type. Above, we have shown that normal law is stable. Very few distribution laws have the property of stability. In the previous   12.6.  Composition of normal laws (Example 2) we made sure that, for example, the law of uniform density is unstable: when we compounded two laws of uniform density in areas from 0 to 1, we obtained Simpson's law.

The stability of a normal law is one of the essential conditions for its wide dissemination in practice. However, the property of stability, in addition to normal, have some other distribution laws. The peculiarity of the normal law is that with a composition of a sufficiently large number of almost arbitrary distribution laws, the total law turns out to be arbitrarily close to normal regardless of what the distribution laws of the terms were. This can be illustrated, for example, by composing the composition of the three laws of uniform density in areas from 0 to 1. The resulting distribution law   12.6.  Composition of normal laws shown in fig. 12.6.1. As can be seen from the drawing, the function graph   12.6.  Composition of normal laws quite reminiscent of the schedule of normal law.

  12.6.  Composition of normal laws

Fig. 12.6.1.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis