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12.7. Linear functions of normally distributed arguments

Lecture



Given a system of random variables   12.7.  Linear functions of normally distributed arguments subordinate to the normal distribution law (or, in short, “formally distributed”); random value   12.7.  Linear functions of normally distributed arguments is a linear function of these quantities:

  12.7.  Linear functions of normally distributed arguments . (12.7.1)

It is required to find the distribution law   12.7.  Linear functions of normally distributed arguments .

It is easy to verify that this is a normal law. Indeed, the magnitude   12.7.  Linear functions of normally distributed arguments is the sum of linear functions, each of which depends on one normally distributed argument   12.7.  Linear functions of normally distributed arguments , and it was proved above that such a linear function is also normally distributed. Adding several normally distributed random variables, we again obtain the value normally distributed.

It remains to find the parameters of   12.7.  Linear functions of normally distributed arguments - center of dispersion   12.7.  Linear functions of normally distributed arguments and standard deviation   12.7.  Linear functions of normally distributed arguments . Applying the theorems on the expectation and variance of a linear function, we obtain:

  12.7.  Linear functions of normally distributed arguments , (12.7.2)

  12.7.  Linear functions of normally distributed arguments , (12.7.3)

Where   12.7.  Linear functions of normally distributed arguments - coefficient of correlation of quantities   12.7.  Linear functions of normally distributed arguments .

In the case where the values   12.7.  Linear functions of normally distributed arguments uncorrelated (and therefore, under normal law, and independent), the formula (12.7.3) takes the form:

  12.7.  Linear functions of normally distributed arguments . (12.7.4)

The standard deviations in formulas (12.7.3) and (12.7.4) can be replaced by proportional probable deviations.

In practice, it is often the case that the laws of the distribution of random variables   12.7.  Linear functions of normally distributed arguments included in the formula (12.7.1) are not exactly known, only their numerical characteristics are known: mathematical expectations and variances. If at the same time values   12.7.  Linear functions of normally distributed arguments independent and their number   12.7.  Linear functions of normally distributed arguments is large enough, so, as a rule, it can be argued that, regardless of the form of the laws of distribution of quantities   12.7.  Linear functions of normally distributed arguments , distribution law   12.7.  Linear functions of normally distributed arguments close to normal. In practice, to obtain a distribution law, which can be approximately taken as normal, it is usually sufficient   12.7.  Linear functions of normally distributed arguments terms in expression (12.7.1). It should be noted that this does not apply to the case when the dispersion of one of the terms in the formula (12.7.1) is overwhelmingly large compared to all the others; it is assumed that random terms in the sum (12.7.1) in their scattering have approximately the same order. If these conditions are met, then for   12.7.  Linear functions of normally distributed arguments a normal law can be approximately adopted with the parameters defined by formulas (12.7.2) and (12.7.4).

Obviously, all the above considerations about the distribution law of a linear function are valid (of course, the approximation is also for the case when the function is not exactly linear, but can be linearized).


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis