13.7. Characteristic functions

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 called function

, (13.7.1)

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

,

functionally related to the value . 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 , it is easy to find its characteristic function.

If a - discontinuous random variable with a number of distribution

then its characteristic function

(13.7.2)

If a - continuous random variable with distribution density then its characteristic function

. (13.7.3)

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

Decision. By the formula (13.7.2) we have:

,

Where .

Example 2. Random variable has a normal distribution:

. (13.7.4)

Determine its characteristic function.

Decision. By the formula (13.7.3) we have:

. (13.7.5)

Using the well-known formula

and bearing in mind that , we get:

. (13.7.6)

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

. (13.7.7)

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

1. If random variables and related by

,

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

. (13.7.8)

Evidence:

.

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 - independent random variables with characteristic functions

and their sum

.

It is required to prove that

. (13.7.9)

We have

.

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

,

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 and with density distribution and . It is required to find the distribution density

.

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

,

after which, subjecting the inverse Fourier transform, find the distribution density :

.

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

with characteristics ; ;

with characteristics , .

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

,

Where ; .

Using the result of example 2, we find

.

According to property 1 of the characteristic functions,

.

Similarly

.

Multiplying and , we have:

,

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