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Normal distribution

Lecture



Normal distribution
Probability density
Normal distribution
The green line corresponds to the standard normal distribution.
Distribution function
Normal distribution
The colors on this graphic correspond to the graphic above.
Designation Normal distribution
Options μ - shift coefficient (real number)
σ > 0 - scale factor (real, strictly positive)
Carrier Normal distribution
Probability density Normal distribution
Distribution function Normal distribution
Expected value Normal distribution
Median Normal distribution
Fashion Normal distribution
Dispersion Normal distribution
Asymmetry coefficient Normal distribution
Coefficient of kurtosis Normal distribution
Informational entropy Normal distribution
Generating function of moments Normal distribution
Characteristic function Normal distribution

The normal distribution , [1] [2] also referred to as the Gauss distribution, is the probability distribution, which in the one-dimensional case is given by a probability density function, the same Gauss function:

Normal distribution

where the parameter μ is the mean (average), the median and the mode of distribution, and the parameter σ is the standard deviation ( σ ² - dispersion) of the distribution.

Thus, the one-dimensional normal distribution is a two-parameter family of distributions. The multidimensional case is described in the article “Multidimensional Normal Distribution”.

The standard normal distribution is called the normal distribution with the expectation μ = 0 and the standard deviation σ = 1.

Content

  • 1 Meaning
  • 2 Properties
    • 2.1 Moments
    • 2.2 Infinite divisibility
    • 2.3 Maximum entropy
  • 3 Simulation of normal pseudorandom variables
  • 4 Normal distribution in nature and applications
  • 5 See also
  • 6 Notes
  • 7 References

Meaning [edit]

The importance of the normal distribution in many areas of science (for example, in mathematical statistics and statistical physics) follows from the central limit theorem of probability theory. If the observation result is the sum of many random weakly interdependent quantities, each of which makes a small contribution relative to the total sum, then with an increase in the number of terms, the distribution of the centered and normalized result tends to normal. This law of probability theory has the consequence of a wide distribution of the normal distribution, which was one of the reasons for its name.

Properties [edit]

Moments [edit]

Moments and absolute moments of a random variable X are called the expectation X p and Normal distribution , respectively. If the expectation of the random variable μ = 0, then these parameters are called central moments . In most cases, moments for p are of interest.

If X has a normal distribution, then for it there exist (finite) moments for all p with a real part greater than −1. For nonnegative integer p , the central moments are as follows:

Normal distribution

Here n is a positive integer, and the record ( p - 1) !! means a double factorial of the number p - 1, that is (since p - 1 is odd in this case) the product of all odd numbers from 1 to p - 1.

The central absolute moments for non-negative integers p are as follows:

Normal distribution

The last formula is also valid for arbitrary p > −1.

Infinite divisibility [edit]

The normal distribution is infinitely divisible.

If random values Normal distribution and Normal distribution independent and have a normal distribution with mathematical expectations Normal distribution and Normal distribution and dispersions Normal distribution and Normal distribution accordingly, Normal distribution also has a normal distribution with the expectation Normal distribution and variance Normal distribution It follows that a normal random variable is representable as the sum of an arbitrary number of independent normal random variables.

Maximum entropy [edit]

The normal distribution is a continuous distribution with maximum entropy for a given expectation and variance [3] [4] .

Simulation of normal pseudo-random variables [edit]

The simplest approximate modeling methods are based on the central limit theorem. Namely, if we add up several independent equally distributed quantities with a finite variance, then the sum will be distributed approximately normally. For example, if we add 100 independent standard uniformly distributed random variables, then the distribution of the sum will be approximately standard normal .

For software generation of normally distributed pseudo-random variables, it is preferable to use the Box-Muller transform. It allows you to generate one normally distributed value on the basis of one evenly distributed.

Normal distribution in nature and applications [edit]

Normal distribution is often found in nature. For example, the following random variables are well modeled by a normal distribution:

  • deflection when shooting.
  • measurement errors (however, the errors of some measuring devices are not normal distributions).
  • some characteristics of living organisms in the population.

Such a wide distribution of this distribution is due to the fact that it is an infinitely divisible continuous distribution with finite variance. Therefore, some others, such as the binomial and Poisson, approach it in the limit. This distribution simulates many non-deterministic physical processes. [five]

The multidimensional normal distribution is used in the study of multidimensional random variables (random vectors). One of the many examples of such applications is the study of the properties of a person’s personality in psychology and psychiatry.

See also [edit]

  • Additive white Gaussian noise
  • Lognormal distribution
  • Central limit theorem
  • Two-dimensional normal distribution
  • Multidimensional normal distribution
  • Statistical criterion
  • Frequency distribution

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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis