Generating function of normal distribution. Normal distribution

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 normally distributed random variable will deviate from the mean by no more than .

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

It can be proven 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 trued more accurately, the greater the number of random variables that are 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 a separate cause, independent of the others . No matter what laws of distribution individual elementary errors are subject to, the features of these distributions in the sum of a large number of terms are leveled out, and the sum turns out to be subject to a law close to normal. The main limitation imposed on the summable errors is that they all uniformly play a relatively small role in the total. If this condition is not met and, for example, one of the random errors turns out to be sharply dominant in its influence on the amount over all others, then the distribution law of this prevailing error will impose its influence on the amount and determine its main features of the distribution law.

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

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

The normal distribution curve has a symmetrical hill-shaped appearance (Fig. 6.1.1). The maximum ordinate of the curve, equal to , corresponds to the point ; As you move away from the point, the distribution density decreases, and at , the curve asymptotically approaches the abscissa.

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

Using variable change

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 famous Euler-Poisson integral:

. (6.1.3)

Hence,

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

Let's calculate the variance of the quantity:

.

Applying the change of variable again

Integrating by parts, we get:

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

Consequently, the parameter in formula (6.1.1) is nothing more than the standard deviation of the value.

Let's find out the meaning of parameters and normal distribution. It is immediately clear from formula (6.1.1) that the center of symmetry of the distribution is the center of dispersion. 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 center of dispersion, the distribution curve will shift along the abscissa axis without changing its shape (Fig. 6.1.2). The center of dispersion characterizes the position of the distribution on the abscissa 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 characteristic of dispersion. The greatest ordinate of the distribution curve is inversely proportional to; as you increase, the maximum ordinate decreases. Since the area of ​​the distribution curve must always remain equal to unity, when increasing, the distribution curve becomes flatter, stretching along the x-axis; on the contrary, with a decrease, the distribution curve stretches upward, simultaneously compressing from the sides, and becomes more needle-shaped. In Fig. 6.1.3 shows three normal curves (I, II, III) at ; of these, curve I corresponds to the largest, 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 decreasing along the other.

The normal distribution law is most often encountered in practice. The main feature that distinguishes it from other laws is that it is a limiting law, to which other laws of distribution approach under very common typical conditions.

Definition. A continuous random variable X has normal law distribution(Gauss's law )with parameters a and σ 2 if its probability density f(x) looks like:

The normal distribution curve is called normal or Gaussian curve. In Fig. 6.5 a), b) shows a normal curve with parameters A And σ 2 and distribution function graph.

Let us pay attention to the fact that the normal curve is symmetrical with respect to the straight line X = A, has a maximum at the point X = A, equal to , and two inflection points X = A σ with ordinates.

It can be noted that in the normal law density expression, the distribution parameters are indicated by the letters A And σ 2, which we used to denote the mathematical expectation and dispersion. This coincidence is not accidental. Let us consider a theorem that establishes the probabilistic theoretical meaning of the parameters of the normal law.

Theorem. The mathematical expectation of a random variable X, distributed according to a normal law, is equal to the parameter a of this distribution, i.e.

and its dispersion – to the parameter σ 2, i.e.

Let's find out how the normal curve will change when the parameters change A And σ .

If σ = const, and the parameter changes a (A 1 < A 2 < A 3), i.e. the center of symmetry of the distribution, then the normal curve will shift along the abscissa axis without changing its shape (Fig. 6.6).

Rice. 6.6

Rice. 6.7

If A= const and the parameter changes σ , then the ordinate of the curve maximum changes f max(a) = . When increasing σ the ordinate of the maximum decreases, but since the area under any distribution curve must remain equal to unity, the curve becomes flatter, stretching along the x-axis. When decreasing σ On the contrary, the normal curve extends upward while simultaneously compressing from the sides (Fig. 6.7).

So the parameter a characterizes the position, and the parameter σ – the shape of a normal curve.

Normal distribution law of a random variable with parameters a= 0 and σ = 1 is called standard or normalized, and the corresponding normal curve is standard or normalized.

The difficulty of directly finding the distribution function of a random variable distributed according to the normal law is due to the fact that the integral of the normal distribution function is not expressed through elementary functions. However, it can be calculated through a special function expressing a definite integral of the expression or. This function is called Laplace function, tables have been compiled for it. There are many varieties of this function, for example:

, .

We will use the function

Let us consider the properties of a random variable distributed according to a normal law.

1. The probability of a random variable X, distributed according to a normal law, falling into the interval [α , β ] equal to

Using this formula, we calculate the probabilities for various values δ (using the table of Laplace function values):

at δ = σ = 2Ф(1) = 0.6827;

at δ = 2σ = 2Ф(2) = 0.9545;

at δ = 3σ = 2Ф(3) = 0.9973.

This leads to the so-called “ three sigma rule»:

If a random variable X has a normal distribution law with parameters a and σ, then it is almost certain that its values ​​lie in the interval(a – 3σ ; a + 3σ ).

Example 6.3. Assuming that the height of men of a certain age group is a normally distributed random variable X with parameters A= 173 and σ 2 = 36, find:

1. Expression of probability density and distribution function of a random variable X;

2. The share of suits of the 4th height (176 - 183 cm) and the share of suits of the 3rd height (170 - 176 cm), which must be included in the total production volume for this age group;

3. Formulate the “three sigma rule” for a random variable X.

1. Finding the probability density

and the distribution function of the random variable X

= .

2. We find the proportion of suits of height 4 (176 – 182 cm) as a probability

R(176 ≤ X ≤ 182) = = Ф(1.5) – Ф(0.5).

According to the table of values ​​of the Laplace function ( Appendix 2) we find:

F(1.5) = 0.4332, F(0.5) = 0.1915.

Finally we get

R(176 ≤ X ≤ 182) = 0,4332 – 0,1915 = 0,2417.

The share of suits of the 3rd height (170 – 176 cm) can be found in a similar way. However, it is easier to do this if we take into account that this interval is symmetrical with respect to the mathematical expectation A= 173, i.e. inequality 170 ≤ X≤ 176 is equivalent to inequality │ X– 173│≤ 3. Then

R(170 ≤X ≤176) = R(│X– 173│≤ 3) = 2Ф(3/6) = 2Ф(0.5) = 2·0.1915 = 0.3830.

3. Let us formulate the “three sigma rule” for the random variable X:

It is almost certain that the height of men in this age group ranges from A – 3σ = 173 – 3 6 = 155 to A + 3σ = 173 + 3·6 = 191, i.e. 155 ≤ X ≤ 191. ◄

7. LIMIT THEOREMS OF PROBABILITY THEORY

As already mentioned when studying random variables, it is impossible to predict in advance what value a random variable will take as a result of a single test - it depends on many reasons that cannot be taken into account.

However, when tests are repeated many times, the behavior of the sum of random variables almost loses its random character and becomes natural. The presence of patterns is associated precisely with the mass nature of phenomena that in their totality generate a random variable that is subject to a well-defined law. The essence of the stability of mass phenomena comes down to the following: the specific features of each individual random phenomenon have almost no effect on the average result of the mass of such phenomena; random deviations from the average, inevitable in each individual phenomenon, are mutually canceled out, leveled out, leveled out in the mass.

It is this stability of averages that represents the physical content of the “law of large numbers,” understood in the broad sense of the word: with a very large number of random phenomena, their result practically ceases to be random and can be predicted with a high degree of certainty.

In the narrow sense of the word, the “law of large numbers” in probability theory is understood as a series of mathematical theorems, each of which, for certain conditions, establishes the fact that the average characteristics of a large number of experiments approach certain certain constants.

The law of large numbers plays an important role in the practical applications of probability theory. The property of random variables, under certain conditions, to behave almost like non-random ones allows one to confidently operate with these quantities and predict the results of mass random phenomena with almost complete certainty.

The possibilities of such predictions in the field of mass random phenomena are further expanded by the presence of another group of limit theorems, which concern not the limiting values ​​of random variables, but the limiting laws of distribution. We are talking about a group of theorems known as the “central limit theorem.” The various forms of the central limit theorem differ from each other in the conditions for which this limiting property of the sum of random variables is established.

Various forms of the law of large numbers with various forms of the central limit theorem form a set of so-called limit theorems probability theory. Limit theorems make it possible not only to make scientific forecasts in the field of random phenomena, but also to evaluate the accuracy of these forecasts.

The law of normal probability distribution of a continuous random variable occupies a special place among various theoretical laws, since it is fundamental in many practical studies. It describes most random phenomena associated with production processes.

Random phenomena that obey the normal distribution law include measurement errors of production parameters, the distribution of technological manufacturing errors, the height and weight of most biological objects, etc.

Normal is the law of probability distribution of a continuous random variable, which is described by a differential function

a - mathematical expectation of a random variable;

Standard deviation of a normal distribution.

The graph of the differential function of the normal distribution is called a normal curve (Gaussian curve) (Fig. 7).

Rice. 7 Gaussian curve

Properties of a normal curve (Gaussian curve):

1. the curve is symmetrical about the straight line x = a;

2. the normal curve is located above the X axis, i.e., for all values ​​of X, the function f(x) is always positive;

3. The ox axis is the horizontal asymptote of the graph, because

4. for x = a, the function f(x) has a maximum equal to

,

at points A and B at and the curve has inflection points whose ordinates are equal.

At the same time, the probability that the absolute value of the deviation of a normally distributed random variable from its mathematical expectation will not exceed the standard deviation is equal to 0.6826.

at points E and G, for and , the value of the function f(x) is equal to

and the probability that the absolute value of the deviation of a normally distributed random variable from its mathematical expectation will not exceed twice the standard deviation is 0.9544.

Asymptotically approaching the x-axis, the Gaussian curve at points C and D, at and , approaches the x-axis very close. At these points the value of the function f(x) is very small

and the probability that the absolute value of the deviation of a normally distributed random variable from its mathematical expectation will not exceed three times the standard deviation is 0.9973. This property of the Gaussian curve is called " three sigma rule".

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

Changing the value of the parameter a (the mathematical expectation of a random variable) does not change the shape of the normal curve, but only leads to its displacement along the X axis: to the right if a increases, and to the left if a decreases.

When a=0, the normal curve is symmetrical about the ordinate.

Changing the value of the parameter (standard deviation) changes the shape of the normal curve: with increasing ordinates of the normal curve they decrease, the curve stretches along the X axis and is pressed against it. As it decreases, the ordinates of the normal curve increase, the curve shrinks along the X axis and becomes more “pointy.”

At the same time, for any values ​​and the area bounded by the normal curve and the X axis remains equal to one (i.e., the probability that a normally distributed random variable will take a value bounded on the X axis of the normal curve is equal to 1).

Normal distribution with arbitrary parameters and , i.e., described by a differential function

called general normal distribution.

The normal distribution with parameters is called normalized distribution(Fig. 8). In a normalized distribution, the differential distribution function is equal to:

Rice. 8 Normalized curve

The cumulative function of the general normal distribution has the form:

Let the random variable X be distributed according to the normal law in the interval (c, d). Then the probability that X will take a value belonging to the interval (c, d) is equal to

Example. The random variable X is distributed according to the normal law. The mathematical expectation and standard deviation of this random variable are equal to a=30 and . Find the probability that X will take a value in the interval (10, 50).

By condition: . Then

Using ready-made Laplace tables (see Appendix 3), we have.

Definition 1

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

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

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

Density of normal distribution.

Let us show that this function is indeed a distribution density. To do this, let's 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 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

The equality is satisfied, which means 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.

Let's consider some simple 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 $x=a$, and $\varphi \left(a\right)=\frac(1)(\sqrt(2\pi )\sigma ) e^(\frac(-((a-a))^2)(2(\sigma )^2))=\frac(1)(\sqrt(2\pi )\sigma )$
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 (Figure 1).

Figure 1. Fig. 1. Normal distribution density graph

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

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

Definition 3

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

Numerical characteristics of 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.

Problem 1: The path length $X$ is a random continuous variable. $X$ is distributed according to the normal distribution law, the mean value of which is equal to $4$ kilometers, and the standard deviation is equal to $100$ 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 imagine all the quantities in one dimension: 100m=0.1km

From Definition 1, we get:

\[\varphi \left(x\right)=\frac(1)(0.1\sqrt(2\pi ))e^(\frac(-((x-4))^2)(0.02 ))\]

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