(real, strictly positive)

Normal distribution, also called Gaussian distribution or Gauss - Laplace- probability distribution , which in the one-dimensional case is given by the probability density function , coinciding with the Gaussian function :

f (x) = 1 σ 2 π e − (x − μ) 2 2 σ 2 , (\displaystyle f(x)=(\frac (1)(\sigma (\sqrt (2\pi ))))\ ;e^(-(\frac ((x-\mu)^(2))(2\sigma ^(2)))),)

where the parameter μ is the mathematical expectation (mean value), median and mode of the distribution, and the parameter σ is the standard deviation ( σ  ² - variance) of the distribution.

Thus, the one-dimensional normal distribution is a two-parameter family of distributions. The multivariate case is described in the article "Multivariate normal distribution".

standard normal distribution is called a normal distribution with mean μ = 0 and standard deviation σ = 1 .

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

  Properties

  Moments

  If random variables X 1 (\displaystyle X_(1)) And X 2 (\displaystyle X_(2)) are independent and have a normal distribution with mathematical expectations μ 1 (\displaystyle \mu _(1)) And μ 2 (\displaystyle \mu _(2)) and dispersions σ 1 2 (\displaystyle \sigma _(1)^(2)) And σ 2 2 (\displaystyle \sigma _(2)^(2)) respectively, then X 1 + X 2 (\displaystyle X_(1)+X_(2)) also has a normal distribution with expected value μ 1 + μ 2 (\displaystyle \mu _(1)+\mu _(2)) and dispersion σ 1 2 + σ 2 2 . (\displaystyle \sigma _(1)^(2)+\sigma _(2)^(2).) This implies that a normal random variable can be represented as the sum of an arbitrary number of independent normal random variables.

  Maximum entropy

  The normal distribution has the maximum differential entropy among all continuous distributions whose variance does not exceed a given value .

  Modeling Normal Pseudo-Random Variables

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

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

  Normal distribution in nature and applications

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

  • shooting deflection.
  • measurement errors (however, the errors of some measuring instruments have non-normal distributions).
  • some characteristics of living organisms in a population.

  This distribution is so widespread because it is an infinitely divisible continuous distribution with finite variance. Therefore, some others approach it in the limit, such as binomial and Poisson. Many non-deterministic physical processes are modeled by this distribution.

  Relationship with other distributions

  • The normal distribution is a type XI Pearson distribution.
  • The ratio of a pair of independent standard normally distributed random variables has a Cauchy distribution. That is, if the random variable X (\displaystyle X) represents the relation X = Y / Z (\displaystyle X=Y/Z)(Where Y (\displaystyle Y) And Z (\displaystyle Z) are independent standard normal random variables), then it will have a Cauchy distribution.
  • If z 1 , … , z k (\displaystyle z_(1),\ldots ,z_(k)) are jointly independent standard normal random variables, i.e. z i ∼ N (0 , 1) (\displaystyle z_(i)\sim N\left(0,1\right)), then the random variable x = z 1 2 + … + z k 2 (\displaystyle x=z_(1)^(2)+\ldots +z_(k)^(2)) has a chi-square distribution with k degrees of freedom.
  • If the random variable X (\displaystyle X) is subject to a lognormal distribution, then its natural logarithm has a normal distribution. That is, if X ∼ L o g N (μ , σ 2) (\displaystyle X\sim \mathrm (LogN) \left(\mu ,\sigma ^(2)\right)), That Y = ln ⁡ (X) ∼ N (μ , σ 2) (\displaystyle Y=\ln \left(X\right)\sim \mathrm (N) \left(\mu ,\sigma ^(2)\right )). And vice versa, if Y ∼ N (μ , σ 2) (\displaystyle Y\sim \mathrm (N) \left(\mu ,\sigma ^(2)\right)), That X = exp ⁡ (Y) ∼ L o g N (μ , σ 2) (\displaystyle X=\exp \left(Y\right)\sim \mathrm (LogN) \left(\mu ,\sigma ^(2) \right)).
  • The ratio of the squares of two standard normal random variables has

  ) plays a particularly important role in probability theory and is most often used in solving practical problems. Its main feature is that it is the limiting law, which is approached by other laws of distribution under very common typical conditions. For example, the sum of a sufficiently large number of independent (or weakly dependent) random variables approximately obeys the normal law, and this is the more accurate, the more random variables are summed.

  It has been experimentally proven that measurement errors, deviations in geometric dimensions and position of elements of building structures during their manufacture and installation, variability of the physical and mechanical characteristics of materials and loads acting on building structures are subject to the normal law.

  Almost all random variables obey the Gaussian distribution, the deviation of which from the average values ​​is caused by a large set of random factors, each of which is individually insignificant (central limit theorem).

  normal distribution called the distribution of a random continuous variable for which the probability density has the form (Fig. 18.1).

  Rice. 18.1. Normal distribution law for a 1< a 2 .

  (18.1)

  where a and are the distribution parameters.

  Probabilistic characteristics random variable, distributed according to the normal law, are equal to:

  Mathematical expectation (18.2)

  Dispersion (18.3)

  Standard deviation (18.4)

  Asymmetry coefficient A = 0(18.5)

  Excess E= 0. (18.6)

  The parameter σ included in the Gaussian distribution is equal to the root-mean-square ratio of a random variable. Value A determines the position of the distribution center (see Fig. 18.1), and the value A- distribution width (Fig. 18.2), i.e. statistical spread around the mean.

  Rice. 18.2. Normal distribution law for σ 1< σ 2 < σ 3

  The probability of falling into a given interval (from x 1 to x 2) for a normal distribution, as in all cases, is determined by the integral of the probability density (18.1), which is not expressed in terms of elementary functions and is represented by a special function, called the Laplace function (integral of probabilities).

  One of the representations of the probability integral:

  Value And called quantile.

  It can be seen that Ф(х) is an odd function, i.e. Ф(-х) = -Ф(х) . The values ​​of this function are calculated and presented in the form of tables in the technical and educational literature.

  
  

  The distribution function of the normal law (Fig. 18.3) can be expressed in terms of the probability integral:

  Rice. 18.2. The function of the normal distribution law.

  The probability that a random variable distributed according to the normal law falls into the interval from X. to x, is determined by the expression:

  It should be noted that

  Ф(0) = 0; Ф(∞) = 0.5; Ф(-∞) = -0.5.

  When solving practical problems related to distribution, one often has to consider the probability of falling into an interval that is symmetric with respect to the mathematical expectation, if the length of this interval i.e. if the interval itself has a boundary from to , we have:

  When solving practical problems, the boundaries of deviations of random variables are expressed through the standard, the standard deviation, multiplied by a certain factor that determines the boundaries of the area of ​​deviations of a random variable.

  Taking and and also using the formula (18.10) and the table F (x) (Appendix No. 1), we obtain

  These formulas show that if a random variable has a normal distribution, then the probability of its deviation from its mean value by no more than σ is 68.27%, by no more than 2σ - 95.45%, and by no more than 3σ - 99.73%.

  Since the value of 0.9973 is close to unity, it is practically considered impossible that the normal distribution of a random variable deviates from the mathematical expectation by more than 3σ. This rule, which is valid only for a normal distribution, is called the three sigma rule. Violation of it is likely P = 1 - 0.9973 = 0.0027. This rule is used when setting the boundaries of permissible deviations of tolerances of geometric characteristics of products and structures.

  Definition 1

  A random variable $X$ has a normal distribution (Gaussian distribution) if the density of its distribution is determined by the formula:

  \[\varphi \left(x\right)=\frac(1)(\sqrt(2\pi )\sigma )e^(\frac(-((x-a))^2)(2(\sigma )^ 2))\]

  Here $aϵR$ is the mathematical expectation, and $\sigma >0$ is the standard deviation.

  Density of the normal distribution.

  Let us show that this function is indeed a distribution density. To do this, check the following condition:

  Consider the improper integral $\int\limits^(+\infty )_(-\infty )(\frac(1)(\sqrt(2\pi )\sigma )e^(\frac(-((x-a))^ 2)(2(\sigma )^2))dx)$.

  Let's make the substitution: $\frac(x-a)(\sigma )=t,\ x=\sigma t+a,\ dx=\sigma dt$.

  Since $f\left(t\right)=e^(\frac(-t^2)(2))$ is an even function, then

  The equality holds, so the function $\varphi \left(x\right)=\frac(1)(\sqrt(2\pi )\sigma )e^(\frac(-((x-a))^2)(2 (\sigma )^2))$ is indeed the distribution density of some random variable.

  Consider some of the simplest properties of the probability density function of the normal distribution $\varphi \left(x\right)$:

  1. The graph of the probability density function of the normal distribution is symmetrical with respect to the straight line $x=a$.
  2. The function $\varphi \left(x\right)$ reaches its maximum at $x=a$, while $\varphi \left(a\right)=\frac(1)(\sqrt(2\pi )\sigma ) e^(\frac(-((a-a))^2)(2(\sigma )^2))=\frac(1)(\sqrt(2\pi )\sigma )$
  3. The function $\varphi \left(x\right)$ decreases as $x>a$ and increases as $x
  4. The function $\varphi \left(x\right)$ has inflection points at $x=a+\sigma $ and $x=a-\sigma $.
  5. The function $\varphi \left(x\right)$ asymptotically approaches the $Ox$ axis as $x\to \pm \infty $.
  6. The schematic graph looks like this (Fig. 1).

  Figure 1 1. Normal distribution density plot

  Note that if $a=0$, then the graph of the function is symmetrical with respect to the $Oy$ axis. Hence the function $\varphi \left(x\right)$ is even.

  Probability normal distribution function.

  To find the probability distribution function for a normal distribution, we use the following formula:

  Hence,

  Definition 2

  The function $F(x)$ is called the standard normal distribution if $a=0,\ \sigma =1$, that is:

  Here $Ф\left(x\right)=\frac(1)(\sqrt(2\pi ))\int\limits^x_0(e^(\frac(-t^2)(2))dt)$ is the Laplace function.

  Definition 3

  Function $Ф\left(x\right)=\frac(1)(\sqrt(2\pi ))\int\limits^x_0(e^(\frac(-t^2)(2))dt)$ is called the probability integral.

  Numerical characteristics of the normal distribution.

  Mathematical expectation: $M\left(X\right)=a$.

  Variance : $D\left(X\right)=(\sigma )^2$.

  Mean square distribution: $\sigma \left(X\right)=\sigma $.

  Example 1

  An example of solving a problem on the concept of normal distribution.

  Task 1: Path length $X$ is random continuous value. $X$ is distributed according to the normal distribution law, the average value of which is $4$ kilometers, and the standard deviation is $100$ meters.

  1. Find the distribution density function $X$.
  2. Construct a diagrammatic plot of the distribution density.
  3. Find the distribution function of the random variable $X$.
  4. Find the variance.
  1. To begin with, let's imagine all the quantities in one dimension: 100m = 0.1km

  From Definition 1, we get:

  \[\varphi \left(x\right)=\frac(1)(0,1\sqrt(2\pi ))e^(\frac(-((x-4))^2)(0,02 ))\]

  (because $a=4\ km,\ \sigma =0.1\ km)$

  1. Using the properties of the distribution density function, we have that the graph of the function $\varphi \left(x\right)$ is symmetric with respect to the straight line $x=4$.

  The function reaches its maximum at the point $\left(a,\frac(1)(\sqrt(2\pi )\sigma )\right)=(4,\ \frac(1)(0,1\sqrt(2\pi )))$

  The schematic graph looks like:

  

  Figure 2.

  1. By definition of the distribution function $F\left(x\right)=\frac(1)(\sqrt(2\pi )\sigma )\int\limits^x_(-\infty )(e^(\frac(-( (t-a))^2)(2(\sigma )^2))dt)$, we have:
  \
  1. $D\left(X\right)=(\sigma )^2=0.01$.

  The normal distribution law (often called the Gauss law) plays an extremely important role in probability theory and occupies a special position among other distribution laws. This is the most common distribution law in practice. The main feature that distinguishes the normal law from other laws is that it is the limiting law, to which other laws of distribution approach under very often encountered typical conditions.

  It can be proved that the sum of a sufficiently large number of independent (or weakly dependent) random variables subject to arbitrary distribution laws (subject to certain very loose restrictions) approximately obeys the normal law, and this is true the more accurately the greater the number of random variables is summed. Most of the random variables encountered in practice, such as, for example, measurement errors, shooting errors, etc., can be represented as the sum of a very large number of relatively small terms - elementary errors, each of which is caused by the action of a separate cause that does not depend on the others . Whatever laws of distribution may be subject to individual elementary errors, the features of these distributions in the sum of a large number of terms are leveled, and the sum turns out to be subject to a law close to normal. The main restriction imposed on summable errors is that they all equally play a relatively small role in the total sum. If this condition is not met, and, for example, one of the random errors turns out to be sharply prevailing over all others in its influence on the sum, then the law of distribution of this prevailing error will impose its influence on the sum and determine in its main features its distribution law.

  Theorems establishing the normal law as the limit for the sum of independent uniformly small random terms will be considered in more detail in Chapter 13.

  The normal distribution law is characterized by a probability density of the form:

  The distribution curve according to the normal law has a symmetrical hilly appearance (Fig. 6.1.1). The maximum ordinate of the curve, equal to , corresponds to the point ; as we move away from the point, the distribution density decreases, and at , the curve asymptotically approaches the abscissa axis.

  

  Let us find out the meaning of the numerical parameters and included in the expression of the normal law (6.1.1); we will prove that the value is nothing but the mathematical expectation, and the value is the standard deviation of the value . To do this, we calculate the main numerical characteristics of the quantity - the mathematical expectation and variance.

  

  Applying the change of variable

  It is easy to verify that the first of the two intervals in formula (6.1.2) is equal to zero; the second is the well-known Euler-Poisson integral:

  . (6.1.3)

  Hence,

  those. the parameter is the mathematical expectation of the value . This parameter, especially in shooting tasks, is often called the center of dispersion (abbreviated as c.r.).

  Let us calculate the dispersion of the quantity :

  .

  Applying again the change of variable

  Integrating by parts, we get:

  The first term in curly brackets is equal to zero (since when decreases faster than any power increases), the second term according to formula (6.1.3) is equal to , whence

  Therefore, the parameter in formula (6.1.1) is nothing but the standard deviation of the value .

  Let us find out the meaning of the parameters and the normal distribution. It can be seen directly from formula (6.1.1) that the center of symmetry of the distribution is the scattering center. This is clear from the fact that when the sign of the difference is reversed, expression (6.1.1) does not change. If you change the dispersion center , the distribution curve will shift along the x-axis without changing its shape (Fig. 6.1.2). The center of scattering characterizes the position of the distribution on the x-axis.

  

  The dimension of the scattering center is the same as the dimension of the random variable .

  The parameter characterizes not the position, but the very shape of the distribution curve. This is the dispersion characteristic. The largest ordinate of the distribution curve is inversely proportional to ; when increasing, the maximum ordinate decreases. Since the area of ​​the distribution curve must always remain equal to unity, as the distribution curve increases, it becomes flatter, stretching along the x-axis; on the contrary, with a decrease, the distribution curve stretches upwards, simultaneously shrinking from the sides, and becomes more needle-like. On fig. 6.1.3 shows three normal curves (I, II, III) at ; of these, curve I corresponds to the largest value, and curve III to the smallest value. Changing the parameter is equivalent to changing the scale of the distribution curve - increasing the scale along one axis and the same decrease along the other.