What is the normal distribution law. Normal distribution law of random variables
Brief theory
Normal is the probability distribution of a continuous random variable whose density has the form:
where is the mathematical expectation and is the standard deviation.
Probability that it will take a value belonging to the interval:
where is the Laplace function:
The probability that the absolute value of the deviation is less than a positive number:
In particular, when the equality holds:
When solving problems that practice poses, one has to deal with various distributions of continuous random variables.
In addition to the normal distribution, the basic laws of distribution of continuous random variables:
Example of problem solution
A part is made on a machine. Its length is a random variable distributed according to a normal law with parameters , . Find the probability that the length of the part will be between 22 and 24.2 cm. What deviation of the length of the part from can be guaranteed with a probability of 0.92; 0.98? Within what limits, symmetrical with respect to , will almost all dimensions of the parts lie?
Solution:
The probability that a random variable distributed according to a normal law will be in the interval:
We get:
The probability that a random variable distributed according to a normal law will deviate from the average by no more than:
By condition
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The most famous and frequently used law in probability theory is the normal distribution law or Gauss's law .
main feature The normal distribution law is that it is a limiting law for other distribution laws.
Note that for a normal distribution the integral function has the form:
.
Let us now show that the probabilistic meaning of the parameters is as follows: A is the mathematical expectation, - the standard deviation (i.e.) of the normal distribution:
a) by definition of the mathematical expectation of a continuous random variable we have
Really
,
since under the integral sign there is an odd function, and the limits of integration are symmetrical with respect to the origin;
- Poisson integral .
So, the mathematical expectation of a normal distribution is equal to the parameter A .
b) by definition of the variance of a continuous random variable and, taking into account that , we can write
.
Integrating by parts, putting 
, let's find
Hence 
.
So, the standard deviation of the normal distribution is equal to the parameter.
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If the distribution is also normal, it is called a normalized (or standard normal) distribution. Then, obviously, the normalized density (differential) and the normalized integral distribution function will be written respectively in the form:
(The function, as you know, is called the Laplace function (see LECTURE 5) or the probability integral. Both functions, that is 
, tabulated and their values are recorded in the corresponding tables).
Properties of normal distribution (properties of normal curve):
1. Obviously, a function on the entire number line.
2. 
, that is, the normal curve is located above the axis Oh
.
3. 
, that is, the axis Oh
serves as the horizontal asymptote of the graph.
4. A normal curve is symmetrical about a straight line x = a (accordingly, the graph of the function is symmetrical about the axis OU ).
Therefore, we can write: .
5. 
.
6. It is easy to show that the points 
And 
are inflection points of the normal curve (prove it yourself).
7.It's obvious that
but since 
, That 
. Besides 
, therefore, all odd moments are equal to zero.
For even moments we can write:
8. 
.
9. 
.
10. 
, Where .
11. For negative values of the random variable: , where .
13. The probability of a random variable falling into a section symmetrical with respect to the center of the distribution is equal to:
EXAMPLE 3. Show that a normally distributed random variable X deviates from mathematical expectation M(X) no more than .
Solution. For normal distribution: 
.
In other words, the probability that the absolute value of the deviation will exceed triple the standard deviation is very small, namely equal to 0.0027. This means that only in 0.27% of cases this can happen. Such events, based on the principle of the impossibility of unlikely events, can be considered practically impossible.
So, an event with a probability of 0.9973 can be considered practically reliable, that is, the random variable deviates from the mathematical expectation by no more than .
EXAMPLE 4. Knowing the characteristics of the normal distribution of a random variable X - tensile strength of steel: kg/mm 2 and kg/mm 2, find the probability of obtaining steel with a tensile strength from 31 kg/mm 2 to 35 kg/mm 2.
Solution.
3. Exponential distribution (exponential distribution law)
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Exponential is the probability distribution of a continuous random variable. X , which is described by a differential function (distribution density)
where is a constant positive value.
The exponential distribution is defined one parameter. This feature of the exponential distribution indicates its advantage compared to distributions that depend on a larger number of parameters. Usually the parameters are unknown and their estimates (approximate values) have to be found; Of course, it is easier to evaluate one parameter than two, or three, etc.
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It is easy to write the integral exponential distribution function:
We defined the exponential distribution using a differential function; it is clear that it can be determined using the integral function.
Comment: Consider a continuous random variable T
- length of time of non-failure operation of the product. Its accepted values are denoted by t
, . Cumulative distribution function 
defines probability of failure products over a period of time t
. Consequently, the probability of failure-free operation during the same time, duration t
, that is, the probability of the opposite event is equal to
Definition. Normal is the probability distribution of a continuous random variable, which is described by the probability density
The normal distribution law is also called Gauss's law.
The normal distribution law occupies a central place in probability theory. This is due to the fact that this law manifests itself in all cases where a random variable is the result of the action of a large number of different factors. All other distribution laws approach the normal law.
It can be easily shown that the parameters and included in the distribution density are, respectively, the mathematical expectation and the standard deviation of the random variable X.
Let's find the distribution function F(x).
The density graph of a normal distribution is called normal curve or Gaussian curve.
A normal curve has the following properties:
1) The function is defined on the entire number line.
2) In front of everyone X the distribution function takes only positive values.
3) The OX axis is the horizontal asymptote of the probability density graph, because with unlimited increase in the absolute value of the argument X, the value of the function tends to zero.
4) Find the extremum of the function.
Because at y’ > 0 at x< m And y'< 0 at x > m, then at the point x = t the function has a maximum equal to .
5) The function is symmetrical with respect to a straight line x = a, because difference
(x – a) is included in the squared distribution density function.
6) To find the inflection points of the graph, we will find the second derivative of the density function.
At x = m+s and x = m- s the second derivative is equal to zero, and when passing through these points it changes sign, i.e. at these points the function has an inflection point.
At these points the value of the function is .
Let's plot the distribution density function.
Graphs were built for T=0 and three possible values of the standard deviation s = 1, s = 2 and s = 7. As you can see, as the value of the standard deviation increases, the graph becomes flatter and the maximum value decreases..
If A> 0, then the graph will shift in a positive direction if A < 0 – в отрицательном.
At A= 0 and s = 1 curve is called normalized. Normalized curve equation: 
For brevity, they say that CB X obeys the law N(m, s), i.e. X ~ N(m, s). The parameters m and s coincide with the main characteristics of the distribution: m = m X, s = s X =. If CB X ~ N(0, 1), then it is called standardized normal value. DF standardized normal value is called Laplace function and is denoted as Ф(x). Using it, you can calculate interval probabilities for the normal distribution N(m, s):
P(x 1 £ X< x 2) = Ф - Ф .
When solving normal distribution problems, it is often necessary to use tabular values of the Laplace function. Since the Laplace function holds the relation F(-x) = 1 - F(x), then it is enough to have table values of the function F(x) only for positive argument values.
For the probability of falling into an interval symmetric with respect to the mathematical expectation, the formula is valid: P(|X - m X |< e) = 2×Ф(e/s) - 1.
The central moments of a normal distribution satisfy the recurrence relation: m n +2 = (n+1)s 2 m n , n = 1, 2, ... . It follows that all central moments of odd order are equal to zero (since m 1 = 0).
Let's find the probability of a random variable distributed according to a normal law falling into a given interval.
Let's denote 
Because the integral is not expressed in terms of elementary functions, then the function is introduced into consideration
,
which is called Laplace function or probability integral.
The values of this function at different meanings X calculated and presented in special tables.
Below is a graph of the Laplace function.
The Laplace function has the following properties:
2) F(- X) = - Ф( X);
The Laplace function is also called error function and denote erf x.
Still in use normalized Laplace function, which is related to the Laplace function by the relation:
Below is a graph of the normalized Laplace function.
When considering the normal distribution law, an important special case stands out, known as three sigma rule.
Let us write down the probability that the deviation of a normally distributed random variable from the mathematical expectation is less than a given value D:
If we take D = 3s, then using tables of values of the Laplace function we obtain:
Those. the probability that a random variable will deviate from its mathematical expectation by an amount greater than triple the standard deviation is practically zero.
This rule is called three sigma rule.
In practice, it is believed that if the three-sigma rule is satisfied for any random variable, then this random variable has a normal distribution.
Example. The train consists of 100 cars. The mass of each car is a random variable distributed according to the normal law with mathematical expectation A= 65 t and standard deviation s = 0.9 t. The locomotive can carry a train weighing no more than 6600 t, otherwise it is necessary to couple a second locomotive. Find the probability that the second locomotive will not be needed.
A second locomotive is not required if the deviation of the train mass from the expected one (100 × 65 = 6500) does not exceed 6600 – 6500 = 100 tons.
Because Since the mass of each car has a normal distribution, then the mass of the entire train will also be normally distributed.
We get:
Example. A normally distributed random variable X is specified by its parameters – a =2 – mathematical expectation and s = 1 – standard deviation. You need to write the probability density and plot it, find the probability that X will take a value from the interval (1; 3), find the probability that X will deviate (in absolute value) from the mathematical expectation by no more than 2.
The distribution density has the form:
Let's build a graph:
Let's find the probability of a random variable falling into the interval (1; 3).
Let us find the probability of deviation of a random variable from the mathematical expectation by an amount not greater than 2.
The same result can be obtained using the normalized Laplace function.
Lecture 8 Law of Large Numbers(Section 2)
Lecture outline
Central limit theorem (general formulation and particular formulation for independent identically distributed random variables).
Chebyshev's inequality.
The law of large numbers in Chebyshev form.
The concept of event frequency.
Statistical understanding of probability.
Law of large numbers in Bernoulli form.
The study of statistical patterns has made it possible to establish that, under certain conditions, the total behavior of a large number of random variables almost loses its random character and becomes natural (in other words, random deviations from some average behavior cancel each other out). In particular, if the influence on the sum of individual terms is uniformly small, the distribution law of the sum approaches normal. The mathematical formulation of this statement is given in a group of theorems called law of large numbers.
LAW OF LARGE NUMBERS – general principle, due to which the joint action of random factors leads, under certain very general conditions, to a result that is almost independent of chance. The first example of the operation of this principle can be the convergence of the frequency of occurrence of a random event with its probability as the number of trials increases (often used in practice, for example, when using the frequency of occurrence of any quality of a respondent in a sample as a sample estimate of the corresponding probability).
Essence law of large numbers is that with a large number of independent experiments, the frequency of occurrence of an event is close to its probability.
Central limit theorem (CLT) (in the formulation of Lyapunov A.M. for identically distributed SVs). If pairwise independent SVs X 1 , X 2 , ..., X n , ... have the same distribution law with finite numerical characteristics M = m and D = s 2 , then for n ® ¥ the distribution law of SVs unlimitedly approaches the normal law N(n×m, ).
Consequence. If in the conditions of the SV theorem 
, then as n ® ¥ the distribution law of SV Y indefinitely approaches the normal law N(m, s/ ).
De Moivre-Laplace theorem. Let SV K be the number of “successes” in n trials according to the Bernoulli scheme. Then, with n ® ¥ and a fixed value of the probability of “success” in one trial p, the distribution law of SV K indefinitely approaches the normal law N(n×p, ).
Consequence. If in the conditions of the theorem, instead of SV K, we consider SV K/n - the frequency of “successes” in n trials according to the Bernoulli scheme, then its distribution law for n ® ¥ and a fixed value of p indefinitely approaches the normal law N(p, ).
Comment. Let SV K be the number of “successes” in n trials according to the Bernoulli scheme. The distribution law of such SV is the binomial law. Then for n ® ¥ the binomial law has two limit distributions:
n distribution Poisson(for n ® ¥ and l = n×p = const);
n distribution Gauss N(n×p, ) (for n ® ¥ and p = const).
Example. The probability of “success” in one trial is only p = 0.8. How many tests must be carried out so that, with a probability of at least 0.9, we can expect that the observed frequency of “success” in tests according to the Bernoulli scheme will deviate from the probability p by no more than e = 0.01?
Solution. For comparison, let's solve the problem in two ways.
In practice, most random variables affected by a large number of random factors are subject to the normal probability distribution law. Therefore, in various applications of probability theory, this law is of particular importance.
The random variable $X$ obeys the normal probability distribution law if its probability distribution density has the following form
$$f\left(x\right)=((1)\over (\sigma \sqrt(2\pi )))e^(-(((\left(x-a\right))^2)\over ( 2(\sigma )^2)))$$
The graph of the function $f\left(x\right)$ is shown schematically in the figure and is called “Gaussian curve”. To the right of this graph is the German 10 mark banknote, which was used before the introduction of the euro. If you look closely, you can see on this banknote the Gaussian curve and its discoverer, the greatest mathematician Carl Friedrich Gauss.
Let's return to our density function $f\left(x\right)$ and give some explanations regarding the distribution parameters $a,\ (\sigma )^2$. The parameter $a$ characterizes the center of dispersion of the values of a random variable, that is, it has the meaning of a mathematical expectation. When the parameter $a$ changes and the parameter $(\sigma )^2$ remains unchanged, we can observe a shift in the graph of the function $f\left(x\right)$ along the abscissa, while the density graph itself does not change its shape.
The parameter $(\sigma )^2$ is the variance and characterizes the shape of the density graph curve $f\left(x\right)$. When changing the parameter $(\sigma )^2$ with the parameter $a$ unchanged, we can observe how the density graph changes its shape, compressing or stretching, without moving along the abscissa axis.
Probability of a normally distributed random variable falling into a given interval
As is known, the probability of a random variable $X$ falling into the interval $\left(\alpha ;\ \beta \right)$ can be calculated $P\left(\alpha< X < \beta \right)=\int^{\beta }_{\alpha }{f\left(x\right)dx}$. Для нормального распределения случайной величины $X$ с параметрами $a,\ \sigma $ справедлива следующая формула:
$$P\left(\alpha< X < \beta \right)=\Phi \left({{\beta -a}\over {\sigma }}\right)-\Phi \left({{\alpha -a}\over {\sigma }}\right)$$
Here the function $\Phi \left(x\right)=((1)\over (\sqrt(2\pi )))\int^x_0(e^(-t^2/2)dt)$ is the Laplace function . The values of this function are taken from . The following properties of the function $\Phi \left(x\right)$ can be noted.
1 . $\Phi \left(-x\right)=-\Phi \left(x\right)$, that is, the function $\Phi \left(x\right)$ is odd.
2 . $\Phi \left(x\right)$ is a monotonically increasing function.
3 . $(\mathop(lim)_(x\to +\infty ) \Phi \left(x\right)\ )=0.5$, $(\mathop(lim)_(x\to -\infty ) \ Phi \left(x\right)\ )=-0.5$.
To calculate the values of the function $\Phi \left(x\right)$, you can also use the function $f_x$ wizard in Excel: $\Phi \left(x\right)=NORMDIST\left(x;0;1;1\right )-0.5$. For example, let's calculate the values of the function $\Phi \left(x\right)$ for $x=2$.
The probability of a normally distributed random variable $X\in N\left(a;\ (\sigma )^2\right)$ falling into an interval symmetric with respect to the mathematical expectation $a$ can be calculated using the formula
$$P\left(\left|X-a\right|< \delta \right)=2\Phi \left({{\delta }\over {\sigma }}\right).$$
Three sigma rule. It is almost certain that a normally distributed random variable $X$ will fall into the interval $\left(a-3\sigma ;a+3\sigma \right)$.
Example 1 . The random variable $X$ is subject to the normal probability distribution law with parameters $a=2,\ \sigma =3$. Find the probability of $X$ falling into the interval $\left(0.5;1\right)$ and the probability of satisfying the inequality $\left|X-a\right|< 0,2$.
Using formula
$$P\left(\alpha< X < \beta \right)=\Phi \left({{\beta -a}\over {\sigma }}\right)-\Phi \left({{\alpha -a}\over {\sigma }}\right),$$
we find $P\left(0.5;1\right)=\Phi \left(((1-2)\over (3))\right)-\Phi \left(((0.5-2)\ over (3))\right)=\Phi \left(-0.33\right)-\Phi \left(-0.5\right)=\Phi \left(0.5\right)-\Phi \ left(0.33\right)=0.191-0.129=$0.062.
$$P\left(\left|X-a\right|< 0,2\right)=2\Phi \left({{\delta }\over {\sigma }}\right)=2\Phi \left({{0,2}\over {3}}\right)=2\Phi \left(0,07\right)=2\cdot 0,028=0,056.$$
Example 2 . Suppose that during the year the price of shares of a certain company is a random variable distributed according to the normal law with a mathematical expectation equal to 50 conventional monetary units and a standard deviation equal to 10. What is the probability that on a randomly selected day of the period under discussion the price for the promotion will be:
a) more than 70 conventional monetary units?
b) below 50 per share?
c) between 45 and 58 conventional monetary units per share?
Let the random variable $X$ be the price of shares of a certain company. By condition, $X$ is subject to a normal distribution with parameters $a=50$ - mathematical expectation, $\sigma =10$ - standard deviation. Probability $P\left(\alpha< X < \beta \right)$ попадания $X$ в интервал $\left(\alpha ,\ \beta \right)$ будем находить по формуле:
$$P\left(\alpha< X < \beta \right)=\Phi \left({{\beta -a}\over {\sigma }}\right)-\Phi \left({{\alpha -a}\over {\sigma }}\right).$$
$$а)\ P\left(X>70\right)=\Phi \left(((\infty -50)\over (10))\right)-\Phi \left(((70-50)\ over (10))\right)=0.5-\Phi \left(2\right)=0.5-0.4772=0.0228.$$
$$b)\P\left(X< 50\right)=\Phi \left({{50-50}\over {10}}\right)-\Phi \left({{-\infty -50}\over {10}}\right)=\Phi \left(0\right)+0,5=0+0,5=0,5.$$
$$in)\ P\left(45< X < 58\right)=\Phi \left({{58-50}\over {10}}\right)-\Phi \left({{45-50}\over {10}}\right)=\Phi \left(0,8\right)-\Phi \left(-0,5\right)=\Phi \left(0,8\right)+\Phi \left(0,5\right)=$$
(real, strictly positive)
Normal distribution, also called Gaussian distribution or Gauss - Laplace- probability distribution, which in the one-dimensional case is specified 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 expectation (mean value), median and mode of the distribution, and the parameter σ is the standard deviation (σ² is the dispersion) 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 mathematical expectation μ = 0 and standard deviation σ = 1.
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The importance of the normal distribution in many fields of science (for example, 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 quantities, 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 be normal. This law of probability theory results in the widespread 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 variances σ 1 2 (\displaystyle \sigma _(1)^(2)) And σ 2 2 (\displaystyle \sigma _(2)^(2)) accordingly, then X 1 + X 2 (\displaystyle X_(1)+X_(2)) also has a normal distribution with mathematical expectation μ 1 + μ 2 (\displaystyle \mu _(1)+\mu _(2)) and variance σ 1 2 + σ 2 2 .(\displaystyle \sigma _(1)^(2)+\sigma _(2)^(2).)
It follows that a normal random variable can be represented as the sum of an arbitrary number of independent normal random variables.
Maximum entropy Normal distribution
has the maximum differential entropy among all continuous distributions whose dispersion does not exceed a given value.
Modeling normal pseudorandom variables The simplest approximate modeling methods are based on the central limit theorem. Namely, if you add several independent identically distributed quantities with finite variance, then the sum will be distributed Fine. For example, if you add 100 independent ones as standard evenly distributed random variables, then the distribution of the sum will be approximately normal.
For programmatic generation of normally distributed pseudorandom variables, it is preferable to use the Box-Muller transformation. It allows you to generate one normally distributed value based on one uniformly distributed value.
Normal distribution in nature and applications
Normal distribution is often found in nature. For example, the following random variables are well modeled by the normal distribution:
- deviation when shooting.
- measurement errors (however, the errors of some measuring instruments are not normally distributed).
- 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, for example, binomial and Poisson. This distribution models many non-deterministic physical processes.
Relationship with other distributions
- The normal distribution is a Pearson type XI 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)- independent standard normal random variables), then it will have a Cauchy distribution.
- If z 1 , … , z k (\displaystyle z_(1),\ldots ,z_(k))- jointly independent standard normal random variables, that is 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 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