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13.7. Characteristic functions

Lecture



One of the most common forms of the central limit theorem was proved by A. M. Lyapunov in 1900. To prove this theorem, A. M. Lyapunov created a special method of characteristic functions. In the future, this method acquired an independent meaning and turned out to be a very powerful and flexible method, suitable for solving various probabilistic problems.

Characteristic function of a random variable   13.7.  Characteristic functions called function

  13.7.  Characteristic functions , (13.7.1)

Where   13.7.  Characteristic functions - imaginary unit. Function   13.7.  Characteristic functions is the mathematical expectation of some complex random variable

  13.7.  Characteristic functions ,

functionally related to the value   13.7.  Characteristic functions . When solving many problems of probability theory, it is more convenient to use characteristic functions than the laws of distribution.

Knowing the distribution law of a random variable   13.7.  Characteristic functions , it is easy to find its characteristic function.

If a   13.7.  Characteristic functions - discontinuous random variable with a number of distribution

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

  13.7.  Characteristic functions

then its characteristic function

  13.7.  Characteristic functions (13.7.2)

If a   13.7.  Characteristic functions - continuous random variable with distribution density   13.7.  Characteristic functions then its characteristic function

  13.7.  Characteristic functions . (13.7.3)

Example 1. Random variable   13.7.  Characteristic functions - number of hits with one shot. The probability of hitting is equal to   13.7.  Characteristic functions . Find the characteristic function of a random variable   13.7.  Characteristic functions .

Decision. By the formula (13.7.2) we have:

  13.7.  Characteristic functions ,

Where   13.7.  Characteristic functions .

Example 2. Random variable   13.7.  Characteristic functions has a normal distribution:

  13.7.  Characteristic functions . (13.7.4)

Determine its characteristic function.

Decision. By the formula (13.7.3) we have:

  13.7.  Characteristic functions . (13.7.5)

Using the well-known formula

  13.7.  Characteristic functions

and bearing in mind that   13.7.  Characteristic functions , we get:

  13.7.  Characteristic functions . (13.7.6)

The formula (13.7.3) expresses the characteristic function   13.7.  Characteristic functions continuous random variable   13.7.  Characteristic functions through its distribution density   13.7.  Characteristic functions . Transformation (13.7.3) to be subjected   13.7.  Characteristic functions , To obtain   13.7.  Characteristic functions called the Fourier transform. Mathematical analysis courses prove that if a function   13.7.  Characteristic functions expressed through   13.7.  Characteristic functions using the Fourier transform, then, in turn, the function   13.7.  Characteristic functions expressed through   13.7.  Characteristic functions using the so-called inverse Fourier transform:

  13.7.  Characteristic functions . (13.7.7)

We formulate and prove the main properties of the characteristic functions.

1. If random variables   13.7.  Characteristic functions and   13.7.  Characteristic functions related by

  13.7.  Characteristic functions ,

Where   13.7.  Characteristic functions - non-random factor, their characteristic functions are related by:

  13.7.  Characteristic functions . (13.7.8)

Evidence:

  13.7.  Characteristic functions .

2. The characteristic function of the sum of independent random variables is equal to the product of the characteristic functions of the terms.

Evidence. Are given   13.7.  Characteristic functions - independent random variables with characteristic functions

  13.7.  Characteristic functions

and their sum

  13.7.  Characteristic functions .

It is required to prove that

  13.7.  Characteristic functions . (13.7.9)

We have

  13.7.  Characteristic functions .

As magnitudes   13.7.  Characteristic functions independent, independent and their functions   13.7.  Characteristic functions . By the theorem of multiplying the mathematical expectations we get:

  13.7.  Characteristic functions ,

Q.E.D.

The apparatus of characteristic functions is often used for the composition of the laws of distribution. Suppose, for example, there are two independent random variables   13.7.  Characteristic functions and   13.7.  Characteristic functions with density distribution   13.7.  Characteristic functions and   13.7.  Characteristic functions . It is required to find the distribution density

  13.7.  Characteristic functions .

This can be done as follows: find the characteristic functions   13.7.  Characteristic functions and   13.7.  Characteristic functions random variables   13.7.  Characteristic functions and   13.7.  Characteristic functions and multiplying them, to obtain the characteristic function of   13.7.  Characteristic functions :

  13.7.  Characteristic functions ,

after which, subjecting   13.7.  Characteristic functions the inverse Fourier transform, find the distribution density   13.7.  Characteristic functions :

  13.7.  Characteristic functions .

Example 3. Using the characteristic functions, find the composition of two normal laws:

  13.7.  Characteristic functions with characteristics   13.7.  Characteristic functions ;   13.7.  Characteristic functions ;

  13.7.  Characteristic functions with characteristics   13.7.  Characteristic functions ,   13.7.  Characteristic functions .

Decision. Find the characteristic function of   13.7.  Characteristic functions . For this we present it in the form

  13.7.  Characteristic functions ,

Where   13.7.  Characteristic functions ;   13.7.  Characteristic functions .

Using the result of example 2, we find

  13.7.  Characteristic functions .

According to property 1 of the characteristic functions,

  13.7.  Characteristic functions .

Similarly

  13.7.  Characteristic functions .

Multiplying   13.7.  Characteristic functions and   13.7.  Characteristic functions , we have:

  13.7.  Characteristic functions ,

and this is the characteristic function of the normal law with parameters   13.7.  Characteristic functions ;   13.7.  Characteristic functions . Thus, the composition of normal laws is obtained by much simpler means than in   13.7.  Characteristic functions 12.6.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis