Density plot and normal distribution functions. Normal (Gaussian) distribution law. Univariate normal distribution

Definition 1

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

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

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

Normal distribution density.

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 (- ((xa)) ^ 2) (2 (\ sigma) ^ 2)) dx) $.

Let's make the replacement: $ \ 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

Equality holds, so the function $ \ varphi \ left (x \ right) = \ frac (1) (\ sqrt (2 \ pi) \ sigma) e ^ (\ frac (- ((xa)) ^ 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 symmetric about the straight line $ x = a $.
2. The $ \ varphi \ left (x \ right) $ function reaches its maximum at $ x = a $, while $ \ varphi \ left (a \ right) = \ frac (1) (\ sqrt (2 \ pi) \ sigma) e ^ (\ frac (- ((aa)) ^ 2) (2 (\ sigma) ^ 2)) = \ frac (1) (\ sqrt (2 \ pi) \ sigma) $
3. The function $ \ varphi \ left (x \ right) $ decreases, for $ x> a $, and increases, for $ x
4. The $ \ varphi \ left (x \ right) $ function 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. Fig. 1. Graph of the density of the normal distribution

Note that if $ a = 0 $, then the graph of the function is symmetric about the $ Oy $ axis. Therefore, the function $ \ varphi \ left (x \ right) $ is even.

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 integral of probability.

Numerical characteristics of the normal distribution.

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 a normal distribution.

Problem 1: The length of the path $ X $ is a random continuous variable. $ 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. Draw a schematic graph of the distribution density.
3. Find the distribution function of the random variable $ X $.
4. Find the variance.

1. To begin with, let's present 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 )) \]

(since $ 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 the definition of the distribution function $ F \ left (x \ right) = \ frac (1) (\ sqrt (2 \ pi) \ sigma) \ int \ limits ^ x _ (- \ infty) (e ^ (\ frac (- ( (ta)) ^ 2) (2 (\ sigma) ^ 2)) dt) $, we have:

\

1. $ D \ left (X \ right) = (\ sigma) ^ 2 = 0.01 $.

The random variable is called distributed according to the normal (Gaussian) law with parameters a and () , if the probability distribution density has the form

A quantity distributed according to the normal law always has an infinite number of possible values, therefore it is convenient to represent it graphically using a distribution density graph. According to the formula

the probability that the random variable will take a value from the interval is equal to the area under the graph of the function on this interval (the geometric meaning definite integral). The function under consideration is non-negative and continuous. The graph of a function has the shape of a bell and is called a Gaussian curve or normal curve.

The figure shows several curves of the distribution density of a random variable given according to the normal law.

All curves have one maximum point, with distance from which to the right and left the curves decrease. The maximum is reached at and is equal to.

Curves are symmetrical about a vertical line drawn through the highest point. The subplot area of ​​each curve is 1.

The difference between the individual distribution curves lies only in the fact that the total area of ​​the subgraph, the same for all curves, is differently distributed between different sections. The main part of the subplot area of ​​any curve is concentrated in the immediate vicinity of the most probable value, and this value is different for all three curves. At different meanings and a different normal laws and different graphs of the density of the distribution function are obtained.

Theoretical studies have shown that most of the random variables encountered in practice have a normal distribution. According to this law, the speed of gas molecules, the weight of newborns, the size of clothes and shoes of the country's population, and many others are distributed. random events physical and biological nature. This pattern was first noticed and theoretically substantiated by A. Moivre.

For, the function coincides with the function, which was already discussed in the local limit theorem of Moivre – Laplace. The probability density of the normal distribution is easy expressed in terms of:

For such values ​​of the parameters, the normal law is called the main .

The distribution function for the normalized density is called Laplace function and denoted Φ (x)... We've also seen this feature before.

Laplace function is independent of specific parameters a and σ. For the Laplace function, using approximate integration methods, tables of values ​​were compiled in the interval with varying degrees accuracy. Obviously, the Laplace function is odd, therefore, there is no need to place its values ​​in the table for negative values.

For a random variable distributed according to the normal law with parameters a and, the mathematical expectation and variance are calculated by the formulas:,. The mean square deviation is equal to.

The probability that a normally distributed quantity will take a value from an interval is equal to

where is the Laplace function introduced in the integral limit theorem.

Often in problems it is required to calculate the probability that the deviation of a normally distributed random variable X of its mathematical expectation in absolute value does not exceed a certain value, i.e. calculate the probability. Applying formula (19.2), we have:

In conclusion, we present one important consequence of formula (19.3). Let's put in this formula. Then, i.e. the probability that the absolute value of the deviation X from its mathematical expectation will not exceed, equal to 99.73%. In practice, such an event can be considered reliable. This is the essence of the Three Sigma Rule.

The Three Sigma Rule. If a random variable is normally distributed, then the absolute value of its deviation from the mathematical expectation practically does not exceed three times the standard deviation.

Definition. Normal is called 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 is central to probability theory. This is due to the fact that this law manifests itself in all cases when a random variable is the result 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 NS.

Find the distribution function F(x) .

The normal distribution density plot is called normal curve or Gaussian curve.

The normal curve has the following properties:

1) The function is defined on the entire number axis.

2) For all NS the distribution function takes only positive values.

3) The OX axis is the horizontal asymptote of the probability density graph, since with an unlimited increase in the absolute value of the argument NS, 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 about a straight line x = a since difference

(x - a) is included in the squared density function.

6) To find the inflection points of the graph, we find the second derivative of the density function.

At x = m+  and x = m-  the second derivative is equal to zero, and when passing through these points it changes sign, i.e. the function has an inflection at these points.

At these points, the value of the function is
.

Let's build a graph of the distribution density function (Fig. 5).

Graphs for T= 0 and three possible values ​​of the standard deviation  = 1,  = 2 and  = 7. As you can see, with an increase in the value of the standard deviation, the graph becomes flatter, and the maximum value decreases.

If a> 0, then the graph will shift in the positive direction if a < 0 – в отрицательном.

At a= 0 and  = 1, the curve is called normalized... Normalized curve equation:

Laplace function

Let us find the probability of a random variable, distributed according to the normal law, falling into a given interval.

We denote

Because integral
is not expressed in terms of elementary functions, then the function is introduced into consideration

,

which is called Laplace function or integral of probabilities.

The values ​​of this function at different values NS calculated and given in special tables.

In fig. 6 shows a graph of the Laplace function.

The Laplace function has the following properties:

1) F (0) = 0;

2) F (-x) = - F (x);

3) F ( ) = 1.

The Laplace function is also called error function and denote erf x.

Still in use normalized the Laplace function, which is related to the Laplace function by the relation:

In fig. 7 shows a graph of the normalized Laplace function.

NS Three sigma rule

When considering the normal distribution law, an important special case is highlighted, known as the 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 :

If we take  = 3, then we obtain using tables of values ​​of the Laplace function:

Those. the probability that a random variable will deviate from its mathematical expectation by more than three times the standard deviation is practically zero.

This rule is called the three sigma rule.

In practice, it is believed that if for any random variable is satisfied rule of three sigma, then this random variable has a normal distribution.

Conclusion on the lecture:

In the lecture, we examined the laws of distribution of continuous quantities. In preparation for the subsequent lecture and practical exercises, you must independently supplement your lecture notes with an in-depth study of the recommended literature and solving the proposed problems.

Normal distribution is the most common type of distribution. One has to meet with him in the analysis of measurement errors, control of technological processes and modes, as well as in the analysis and forecasting of various phenomena in biology, medicine and other fields of knowledge.

The term "normal distribution" is used in a conventional sense as generally accepted in the literature, although not entirely successful. So, the statement that some feature obeys the normal distribution law does not at all mean the presence of any unshakable norms that supposedly underlie the phenomenon, the reflection of which is the feature in question, and obedience to other distribution laws does not mean any abnormality of this phenomenon.

The main feature of the normal distribution is that it is the limiting one, which is approached by other distributions. The normal distribution was first discovered by Moivre in 1733. Only continuous random variables obey the normal law. The density of the normal distribution law has the form.

The mathematical expectation for a normal distribution is equal to. The variance is equal.

Basic properties of the normal distribution.

1. The distribution density function is defined on the entire number axis Oh , that is, each value NS corresponds to a well-defined value of the function.

2. For all values NS (both positive and negative) the density function takes positive values, that is, the normal curve is located above the axis Oh .

3. Limit of the density function with unlimited increase NS is zero,.

4. The normal distribution density function at a point has a maximum.

5. The graph of the density function is symmetrical about a straight line.

6. The distribution curve has two inflection points with coordinates and.

7. Mode and median of normal distribution coincide with mathematical expectation a .

8. The shape of the normal curve does not change when the parameter is changed a .

9. The coefficients of skewness and kurtosis of the normal distribution are equal to zero.

The importance of calculating these coefficients for empirical distribution series is obvious, since they characterize the slope and steepness of the given series in comparison with the normal one.

The probability of hitting the interval is found by the formula, where is an odd tabulated function.

Let us determine the probability that a normally distributed random variable deviates from its mathematical expectation by an amount less, that is, we find the probability of inequality, or the probability of double inequality. Substituting into the formula, we get

Expressing the deviation of a random variable NS in fractions of the standard deviation, that is, putting in the last equality, we get.

Then at we get,

when we get

when we get.

From the last inequality it follows that practically the scattering of a normally distributed random variable is contained in the section. The probability that a random variable will not fall on this area is very small, namely equal to 0.0027, that is, this event can occur only in three cases out of 1000. Such events can be considered practically impossible. The above reasoning is based on the three sigma rule, which is formulated as follows: if a random variable has a normal distribution, then the deviation of this value from the mathematical expectation in absolute value does not exceed three times the standard deviation.

Example 28. A part made by an automatic machine is considered suitable if the deviation of its controlled size from the design size does not exceed 10 mm. Random deviations of the controlled size from the design one are subject to the normal distribution law with standard deviation mm and mathematical expectation. What percentage of usable parts does the machine produce?

Solution. Consider a random variable NS - deviation of the size from the design. The part will be considered good if the random value belongs to the interval. We find the probability of manufacturing a suitable part by the formula. Consequently, the percentage of good parts produced by the automatic machine is 95.44%.

Binomial distribution

Binomial is the probability distribution of occurrence m number of events in NS independent tests, in each of which the probability of the occurrence of an event is constant and equal to R ... The probability of the possible number of occurrences of an event is calculated by the Bernoulli formula:,

where . Permanent NS and R in this expression are the parameters of the binomial law. The binomial distribution describes the probability distribution of a discrete random variable.

Basic numerical characteristics of the binomial distribution. The mathematical expectation is equal. The variance is equal. The skewness and kurtosis coefficients are equal to and. With an unlimited increase in the number of tests A and E tend to zero, therefore, we can assume that the binomial distribution converges to normal with increasing number of trials.

Example 29. Independent tests are performed with the same probability of occurrence of the event A in every trial. Find the probability of an event occurring A in one trial if the variance in the number of occurrences in the three trials is 0.63.

Solution. For binomial distribution. Substitute the values, get from here or then and.

Poisson distribution

Distribution Law of Rare Phenomena

Poisson distribution describes the number of events m occurring at equal intervals of time, provided that the events occur independently of each other with a constant average intensity. Moreover, the number of tests NS is high, and the probability of the occurrence of an event in each test R small. Therefore, the Poisson distribution is called the law of rare phenomena or the simplest flow. The parameter of the Poisson distribution is a quantity characterizing the intensity of the occurrence of events in NS tests. Poisson distribution formula.

The Poisson distribution describes well the number of claims for the payment of insurance premiums per year, the number of calls received at the telephone exchange for a certain time, the number of failures of elements during the reliability test, the number of defective products, and so on.

Basic numerical characteristics for the Poisson distribution. The mathematical expectation is equal to the variance and is equal to a ... That is . This is the hallmark of this distribution. The skewness and kurtosis coefficients are respectively equal.

Example 30. The average number of insurance payments per day is two. Find the probability that in five days you will have to pay: 1) 6 sums insured; 2) less than six amounts; 3) at least six. Distribution.

This distribution is often observed when studying the lifetimes of various devices, the uptime of individual elements, parts of the system and the system as a whole, when considering the random time intervals between the occurrence of two successive rare events.

The exponential distribution density is determined by a parameter called failure rate... This term is associated with a specific area of ​​application - reliability theory.

The expression for the integral function of the exponential distribution can be found using the properties of the differential function:

Exponential distribution, variance, standard deviation. Thus, this distribution is characterized by the fact that the standard deviation is numerically equal to mathematical expectation... For any value of the parameter, the coefficients of asymmetry and kurtosis are constant values.

Example 31. The average TV operation time before the first failure is 500 hours. Find the probability that a randomly taken TV will work more than 1000 hours without breakdowns.

Solution. Since the average operating time to the first failure is 500, then. We find the required probability by the formula.

In the theory of probability, a fairly large number of different distribution laws are considered. For solving problems related to the construction of control charts, only a few of them are of interest. The most important of these is normal distribution, which is used to build control charts used in quantitative control, i.e. when we are dealing with a continuous random variable. The normal distribution law occupies a special position among other distribution laws. This is explained by the fact that, firstly, it is most often encountered in practice, and, secondly, it is a limiting law, to which other distribution laws approach under very often encountered typical conditions. As for the second circumstance, in the theory of probability it is proved that the sum of a sufficiently large number of independent (or weakly dependent) random variables, subject to any distribution laws (subject to some very loose restrictions), approximately obeys the normal law, and this is carried out all the more precisely, the more random variables are added. Most of the random variables encountered in practice, such as measurement errors, can be represented as the sum of a very large number of relatively small terms - elementary errors, each of which is caused by a separate cause independent of the others. The normal law manifests itself in cases where the random variable NS is the result of a large number of different factors. Each factor separately by the value NS influences insignificantly, and it is impossible to indicate which one influences more than others.

Normal distribution(Laplace – Gauss distribution) Is the probability distribution of a continuous random variable NS such that the probability distribution density at - ¥<х< + ¥ принимает действительное значение:

Exp (3)

That is, the normal distribution is characterized by two parameters m and s, where m is the mathematical expectation; s is the standard deviation of the normal distribution.

The quantity s 2 Is the variance of the normal distribution.

The mathematical expectation m characterizes the position of the distribution center, and the standard deviation s (RMS) is a characteristic of dispersion (Fig. 3).

f (x) f (x)

Figure 3 - Density functions of the normal distribution with:

a) different mathematical expectations m; b) different standard deviations s.

Thus, the value μ is determined by the position of the distribution curve on the abscissa axis. Dimension μ - the same as the dimension of the random variable X... With an increase in the mathematical expectation, both functions are shifted in parallel to the right. With decreasing variance s 2 the density concentrates more and more around m, while the distribution function becomes steeper.

The value of σ determines the shape of the distribution curve. Since the area under the distribution curve must always remain equal to one, the distribution curve becomes flatter with increasing σ. In fig. 3.1 shows three curves for different σ: σ1 = 0.5; σ2 = 1.0; σ3 = 2.0.

Figure 3.1 - Density functions of the normal distribution with different standard deviations s.

The distribution function (integral function) has the form (Fig. 4):

(4)

Figure 4 - Integral (a) and differential (b) normal distribution functions

Especially important is the linear transformation of a normally distributed random variable NS, after which a random variable is obtained Z with a mathematical expectation of 0 and a variance of 1. Such a transformation is called normalization:

It can be done for each random variable. Normalization allows all possible variants of the normal distribution to be reduced to one case: m = 0, s = 1.

A normal distribution with m = 0, s = 1 is called normalized normal distribution (standardized).

Standard normal distribution(the standard Laplace – Gauss distribution or normalized normal distribution) is the probability distribution of the standardized normal random variable Z, the distribution density of which is equal to:

at - ¥<z< + ¥

Function values Ф (z) determined by the formula:

(7)

Function values Ф (z) and density f (z) normalized normal distribution calculated and tabulated (tabulated). The table is compiled for positive values ​​only. z therefore:

F (–z) = 1–Ф (z) (8)

Using these tables, it is possible to determine not only the values ​​of the function and the density of the normalized normal distribution for a given z, but also the values ​​of the function of the general normal distribution, since:

; (9)

. 10)

In many problems related to normally distributed random variables, it is necessary to determine the probability of hitting a random variable NS, subject to the normal law with parameters m and s, to a certain section. Such a section can be, for example, the tolerance field for a parameter from the upper value U to the bottom L.

The probability of hitting the interval from NS 1 to NS 2 can be determined by the formula:

Thus, the probability of hitting a random variable (parameter value) NS in the tolerance field is determined by the formula