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Ministry of Education and Science of the Russian Federation

Federal State Budgetary Educational Institution

higher professional education

"South Ural State University

(national research university)"

Faculty of Instrument Engineering (KTUR)

Department of Information and Measurement Technology

Abstract on the topic

“What is a random variable?”

in the discipline "Probability Theory and Mathematical Statistics"

Checked:

______________/ A.P. Lapin

Completed:

student of group PS-236

_______________/Zagoskin Y.S./

Chelyabinsk 2015

INTRODUCTION

1. RANDOM VARIABLE

CONCLUSION

BIBLIOGRAPHICAL LIST

INTRODUCTION

Probability theory is a relatively young, but already classical, branch of mathematics. Its development as a separate science occurred in mid-17th century century, and began with the correspondence of two world-famous French mathematicians: Blaise Pascal and Pierre de Fermat. However, tasks related to calculating probabilities in gambling, scientists began to become interested much earlier. For example, the Italian mathematician Luca Pacioli, back in 1494, in his work “Summa de arithmetica, geometria, proportioni et proportionalitа”), considered one of the problems about probability, but, unfortunately, made an erroneous decision.

Today, methods of probability theory and mathematical statistics are an integral part of almost any discipline, both technical and humanitarian. The laws of distribution of random variables turned out to be applicable not only to mathematics, physics, chemistry, and so on, but also to disciplines that are partly predictive in nature, such as sociology, economics, political science, etc.

In this work, we will get acquainted with the basic concepts, terms and laws of probability theory and mathematical statistics, as well as with the application of the latter in practice.

1. RANDOM VARIABLE

1.1 Definition random variable

A random variable is fundamental concept probability theory and mathematical statistics.

Each author formulates the concept of a random variable in his own way. E.S. Wentzel, for example, defines a random variable as a quantity that, as a result of experiment, can take on one or another value, and it is not known in advance which one.

In other words, a random variable is a value that has a whole set acceptable values, but it accepts only one thing, and which one it is impossible to say for sure in advance.

Formal mathematical definition random variable sounds like this:

Let (Ш, F, P) - probability space, then the function X is called a random variable: Ш > R.

A random variable in practice is usually denoted in capital letters, for example: X, Y, Z, then the possible values of the quantity itself are determined by lowercase characters: x, y, z.

1.2 Types and examples of random variables

There are two types of random variables: discrete and continuous.

Discrete are those random variables whose set of values is finite or fixed. An example of a discrete random variable is the number of hits on the target when a certain number shots.

A continuous random variable is one whose set of values is uncountable or infinite. As an example for a continuous random variable, you can take the number of circles on the water after a stone hits it, or the distance that an arrow flies before falling to the ground.

All random variables, among other things, have one more important characteristic- a range of permissible values, which, in turn, can be either limited or unlimited. Hence, we have, depending on the number of permissible values, limited random variables, the number of permissible values is finite or fixed, and unlimited, the number of permissible values of which is infinite.

Discrete random variables can have a limited and unlimited range of possible values, while continuous ones have only an unlimited range.

In practice, in probability theory and mathematical statistics, as a rule, deal only with continuous random variables.

2. LAWS OF RANDOM VARIABLE DISTRIBUTION

2.1 Law of distribution of a discrete random variable

Any relationship between the permissible values of a random variable and the probabilities of their occurrence is called the distribution law of a discrete random variable.

There are two ways to specify the distribution law:

· Analytically, when the distribution law is specified in the form of a table of correspondence between the values of a random variable and their probability, called the distribution series:

Table 1 - random variable distribution series

Here, the first line contains the possible values of the random variable, and the second line contains their probabilities, with the sum of all probabilities equal to one:

· Graphically, when a random value distribution table accepts a distribution polygon:

Figure 1 - random variable distribution polygon

Where the sum of all ordinates of the polygon is the probability of all admissible values of the random variable, therefore also equal to one.

There is also a binomial law of distribution of a discrete random variable or, secondly, the Bernoulli distribution law.

Definition: a discrete random variable o is distributed according to the binomial law if the probability that event A occurs exactly m times in a series of n trials according to the Bernoulli scheme is equal to:

Or in table form:

Table 2 - series of binomial distribution

An example is selective quality control of production products, in which the selection of products for testing is carried out according to a random repeated sampling scheme, i.e. when inspected items are returned to the original batch. Then the number of non-standard products among the selected ones is a random variable with binomial law probability distributions.

A discrete random variable is called distributed according to Poisson's law if it has an unlimited countable set of permissible values 0, 1, 2, ..., m, ... Then the corresponding probabilities are determined by formula (3):

M = 0, 1, 2,…; (3)

An example of a phenomenon distributed according to Poisson's law is the sequence radioactive decay particles.

2.2 Laws of distribution of a continuous random variable

random variable theory probability

The rules of distribution of a random variable discussed above are valid only in relation to discrete quantities, due to the fact that all of the laws listed are constructed solely from the consideration that the number of possible values of a random variable is finite and strictly fixed. This is why, for example, it will not be possible to distribute a continuous random variable according to the Poisson or Bernoulli law, since it is impossible to list the number of permissible values of a given value - it is infinite.

The following laws exist to describe the distribution of continuous random variables:

Consider the values of the random variable X such that X<х. Вероятность события X<х зависит от x, т.е. является функцией x. Эта функция и называется интегральной функцией распределения и обозначается через F(x):

Equality (4) reads:

The probability that a random value X is to the left of a value x is determined by the distribution function F(x).

Figure 2 - Graphical representation of the r.v. distribution function.

It is worth noting that in the form of a distribution function, it is possible to describe both continuous and discrete random variables - this is a universal characteristic.

For continuous random variables in practice, along with the distribution function F(x), it is also customary to use another distribution law - the probability distribution density of the random variable:

Equality (5) is the differential distribution law of a random variable, which expresses the slope of the distribution function F(x).

Figure 3 - Graphical representation of the differential distribution law of r.v.

Note that the differential law of distribution of a random variable is not universal - it is applicable exclusively to continuous random variables.

One of the laws often used in practice is the normal distribution law - the Gaussian distribution law. The law characterizes the probability density of a normally distributed random variable X and has the form:

Where a and y distribution parameters have the following values:

The distribution curve (Figure 4a), or Gaussian curve, is obtained symmetrically relative to the point x = a - the maximum point. As the value of y decreases, the ordinate of the maximum point increases infinitely, while the curve diverges proportionally along the abscissa axis, keeping the area of the graph constant, equal to one (Figure 4b).

Figure 4 - Distribution curves:

4a - Gaussian curve,

4b - behavior of the Gaussian curve when parameter y changes;

In practice, the normal distribution plays a significant role in many fields of knowledge, but special attention is paid to it in physics. A physical quantity obeys Gauss's law when it is subject to a large number of random disturbances, which is an extremely common situation, as a result of which the normal distribution is most often found in nature, and this is where its name comes from.

A continuous random variable is called uniformly distributed on the interval (a, b) if all its possible values belong to this interval and the probability distribution density is constant - the law of uniform distribution of a continuous random variable, which has the form:

For a random variable X uniformly distributed in the interval (a, b) (Figure 5), the probability of falling into any interval (x1, x2) lying inside the interval (a, b) is equal to:

Figure 5 - Uniform distribution density graph

As an example of uniformly distributed quantities, we can take rounding errors. So, if all tabular values of a certain function are rounded to the same digit, then choosing a tabular value at random, we consider that the rounding error of the selected number is a random variable uniformly distributed in the interval where.

A continuous random variable X is called exponentially distributed if its probability distribution density has the form:

As an example, let's take the failure-free operation time T of a computer system, where T is a random variable that has an exponential distribution with parameter l, the physical meaning of which is the average number of failures per unit time, not counting system downtime for repairs.

Figure 6 - Density graph of exponential distribution

CONCLUSION

Methods, tools and laws of probability theory and mathematical statistics throughout all stages of the formation of the discipline were relevant, as they remain to this day. The main principle of the methods, which made it possible to touch upon such a huge number of industries and areas of knowledge, is universality. They can be easily applied in any discipline, and at the same time they do not lose their power and remain fair.

But never before has probability theory been as in demand as it is today. This is primarily due to the incredible pace of development and growth of computer technology. Every year it becomes more and more complex, performance increases, the number of operations performed per second increases, and all this happens not without the participation of mathematical statistics, which, in its turn, helps to optimize the operation of computer systems and complexes, increases the accuracy of calculations, and carries out a predictive function.

This work partially helps to understand the basics of the discipline. Introduces fundamental concepts such as discrete and continuous random variables and explains the difference between the latter. Introduces the laws of their distribution, with the further application of all acquired knowledge in practice.

BIBLIOGRAPHICAL LIST

1. Ventzel, E.S. Probability theory / E.S. Ventzel - M.: Nauka, 1969.

2. Smirnov, N.V. Course on probability theory and mathematical statistics for technical applications./ N.V. Smirnov, I.V. Dunin-Barkovsky - M.: “Science”, 1969.

3. Pustylnik, E.I. Statistical methods of analysis and processing of observations: textbook / E.I. Leonurus. - M.: “Science”, 1968.

4. Johnson, N. Statistics and planning in science and technology. / N. Johnson, F. Lyon - M.: “Mir”, 1969.

5.http://www.wikipedia.org/

Annotation

Zagoskin Y.S. “What is a random variable?”

Chelyabinsk: Yuurgu

Bibliography List - 5 names.

Purpose of the abstract: To become familiar with the basic terms of probability theory and mathematical statistics.

Abstract objectives: Understand the concept of a random variable.

The concept of a random variable is considered, the classification of random variables is determined, the laws of their distribution are considered, examples of the application of laws and methods in practice, and the prospects of the discipline are analyzed.

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The concept of a random variable

Random is a quantity that, as a result of testing, takes on one or another (but only one) possible value, unknown in advance, varying from test to test and depending on random circumstances. Unlike a random event, which is a qualitative characteristic of a random test result, a random variable characterizes the test result quantitatively. Examples of a random variable include the size of the workpiece, the error in the measurement result of any parameter of a product or environment. Among the random variables encountered in practice, two main types can be distinguished: discrete and continuous.

Discrete is a random variable that takes a finite or infinite countable set of values. For example: hit rate with three shots; number of defective products in a batch of n pieces; the number of calls received at the telephone exchange during the day; the number of failures of device elements over a certain period of time when testing it for reliability; number of shots until the first hit on the target, etc.

Continuous is a random variable that can take any value from some finite or infinite interval. Obviously, the number of possible values of a continuous random variable is infinite. For example: error when measuring the radar range; uptime of the microcircuit; manufacturing error of parts; salt concentration in sea water, etc.

Random variables are usually denoted by the letters X,Y, etc., and their possible values by x,y, etc. To define a random variable, it is not enough to list all its possible values. It is also necessary to know how often certain of its values can appear as a result of tests under the same conditions, i.e., you need to set the probabilities of their occurrence. The set of all possible values of a random variable and their corresponding probabilities constitutes the distribution of the random variable.

Laws of random variable distribution

Law of distribution A random variable is the correspondence between possible values of a random variable and their corresponding probabilities. A random variable is said to obey a given distribution law. Two random variables are called independent, if the distribution law of one of them does not depend on what possible values the other quantity took. Otherwise, the random variables are called dependent. Several random variables are called mutually independent, if the laws of distribution of any number of them do not depend on what possible values the other quantities took.

The distribution law of a random variable can be specified in the form of a table, distribution function or distribution density. A table containing possible values of a random variable and the corresponding probabilities is the simplest form of specifying the distribution law of a random variable.

\begin(array)(|c|c|c|c|c|c|c|)\hline(X)&x_1&x_2&x_3&\cdots&x_(n-1)&x_n\\\hline(P)&p_1&p_2&p_3&\cdots&p_(n-1 )&p_n\\\hline\end(array)

The tabular definition of the distribution law can only be used for a discrete random variable with a finite number of possible values. The tabular form of specifying the law of a random variable is also called a distribution series.

For clarity, the distribution series is presented graphically. When displayed graphically in a rectangular coordinate system, all possible values of a random variable are plotted along the abscissa axis, and the corresponding probabilities are plotted along the ordinate axis. Points (x_i,p_i) connected by straight line segments are called distribution polygon(Fig. 5). It should be remembered that connecting the points (x_i,p_i) is performed only for the purpose of clarity, since in the intervals between x_1 and x_2, x_2 and x_3, etc. there are no values that the random variable X can take, so the probability of its appearance in these intervals are equal to zero.

A distribution polygon, like a distribution series, is one of the forms of specifying the distribution law of a discrete random variable. They can have different shapes, but they all have one common property: the sum of the ordinates of the vertices of the distribution polygon, which is the sum of the probabilities of all possible values of the random variable, is always equal to one. This property follows from the fact that all possible values of the random variable X form a complete group of incompatible events, the sum of the probabilities of which is equal to one.

Probability distribution function and its properties

The distribution function is the most general form of specifying the distribution law. It is used to specify both discrete and continuous random variables. It is usually denoted F(x) . Distribution function determines the probability that a random variable X takes values less than a fixed real number x, i.e. F(x)=P\(X cumulative distribution function.

The geometric interpretation of the distribution function is very simple. If a random variable is considered as a random point X of the Ox axis (Fig. 6), which as a result of the test can take one or another position on the axis, then the distribution function F(x) is the probability that the random point X as a result of the test will fall to the left points x.

For a discrete random variable X, which can take values , the distribution function has the form

F(x)=\sum\limits_(x_i
where inequality x_i

A continuous random variable has a continuous distribution function; the graph of this function has the shape of a smooth curve (Fig. 8).

Let's consider the general properties of distribution functions.

Property 1. The distribution function is non-negative, a function between zero and one:

0\leqslant(F(x))\leqslant1

The validity of this property follows from the fact that the distribution function F(x) is defined as the probability of a random event consisting in the fact that X

Property 2. The probability of a random variable falling into the interval [\alpha;\beta) is equal to the difference between the values of the distribution function at the ends of this interval, i.e.

P\(\alpha\leqslant(X)<\beta\}=F(\beta)-F(\alpha)

It follows that the probability of any individual value of a continuous random variable is zero.

Property 3. The distribution function of a random variable is a non-decreasing function, i.e. F(\beta)\geqslant(F(\alpha)).

Property 4. At minus infinity the distribution function is equal to zero, and at plus infinity it is equal to one, i.e. \lim_(x\to-\infty)F(x)=0 And \lim_(x\to+\infty)F(x)=1.

Example 1. The distribution function of a continuous random variable is given by the expression

F(x)=\begin(cases)0,&x\leqslant1\\a(x-1)^2,&1 0\end(cases).

Find coefficient a and plot F(x) . Determine the probability that the random variable X will take a value on the interval as a result of the experiment.

Solution. Since the distribution function of a continuous random variable X is continuous, then for x=3 we obtain a(3-1)^2=1. Hence a=\frac(1)(4) . The graph of the function F(x) is shown in Fig. 9.

Based on the second property of the distribution function, we have

P\(1\leqslant(X)<2\}=F(2)-F(1)=\frac{1}{4}.

Probability distribution density and its properties

The distribution function of a continuous random variable is its probabilistic characteristic. But it has the disadvantage that it is difficult to judge from it the nature of the distribution of a random variable in a small neighborhood of one or another point on the numerical axis. A more clear idea of the nature of the distribution of a continuous random variable is given by a function called the probability distribution density, or differential distribution function of a random variable.

Distribution density f(x) is equal to the derivative of the distribution function F(x), i.e.

F(x)=F"(x).

The meaning of the distribution density f(x) is that it indicates how often a random variable X appears in some neighborhood of a point x when repeating experiments. A curve depicting the distribution density f(x) of a random variable is called distribution curve.

Let's consider distribution density properties.

Property 1. The distribution density is non-negative, i.e.

F(x)\geqslant0.

Property 2. The distribution function of a random variable is equal to the integral of the density in the interval from -\infty to x, i.e.

F(x)=\int\limits_(-\infty)^(x)f(x)\,dx.

Property 3. The probability of a continuous random variable X falling into a section (\alpha;\beta) is equal to the integral of the distribution density taken over this section, i.e.

P\(\alpha\leqslant(X)\leqslant\beta\)=\int\limits_(\alpha)^(\beta)f(x)\,dx.

Property 4. The integral over infinite limits of the distribution density is equal to unity:

\int\limits_(-\infty)^(+\infty)f(x)\,dx=1.

Example 2. Random variable X is subject to a distribution law with density

F(x)=\begin(cases)0,&x<0\\a\sin{x},&0\pi\end(cases)

Determine coefficient a; construct a distribution density graph; find the probability of a random variable falling into the area from 0 to \frac(\pi)(2), determine the distribution function and construct its graph.

\int\limits_(-\infty)^(+\infty)f(x)\,dx=a\int\limits_(0)^(\pi)\sin(x)\,dx=\Bigl.(- a\cos(x))\Bigl|_(0)^(\pi)=2a.

Taking into account property 4 of the distribution density, we find a=\frac(1)(2) . Therefore, the distribution density can be expressed as follows:

F(x)=\begin(cases)0,&x<0\\\dfrac{1}{2}\sin{x},&0\pi\end(cases).

Distribution density graph in Fig. 10. By property 3, we have

P\!\left\(0

To determine the distribution function, we use property 2:

F(x)=\frac(1)(2)\int\limits_(0)^(x)\sin(x)\,dx=\Bigl.(\-\frac(1)(2)\cos( x))\Bigl|_(0)^(x)=\frac(1)(2)-\frac(1)(2)\cos(x).

Thus we have

F(x)=\begin(cases)0,&x<0\\\dfrac{1}{2}-\dfrac{1}{2}\cos{x},&0\pi\end(cases).

The distribution function graph is shown in Fig. 11

Numerical characteristics of random variables

The distribution law fully characterizes a random variable from a probabilistic point of view. But when solving a number of practical problems, there is no need to know all possible values of a random variable and the corresponding probabilities, but it is more convenient to use some quantitative indicators. Such indicators are called numerical characteristics of a random variable. The main ones are mathematical expectation, dispersion, moments of various orders, mode and median.

The mathematical expectation is sometimes called the average value of a random variable. Consider a discrete random variable X taking the values x_1,x_2,\ldots,x_n with probabilities accordingly p_1,p_2,\ldots,p_n Let us determine the arithmetic mean of the values of a random variable, weighted by the probabilities of their occurrence. Thus, we calculate the average value of a random variable, or its mathematical expectation M(X) :

M(X)=\frac(x_1p_1+x_2p_2+\cdots+x_np_n)(p_1+p_2+\cdots+p_n)=\frac(\sum\limits_(i=1)^(n)x_ip_i)(\sum\limits_( i=1)^(n)p_i).

Considering that \sum\limits_(i=1)^(n)p_i=1 we get

M(X)=\sum\limits_(i=1)^(n)x_ip_i).~~~~~~~(4.1)

So, mathematical expectation A discrete random variable is the sum of the products of all its possible values and the corresponding probabilities.

For a continuous random variable, the mathematical expectation

M(X)=\int\limits_(-\infty)^(\infty)xf(x)\,dx.

Expectation of a continuous random variable X, the possible values of which belong to the segment,

M(X)=\int\limits_(a)^(b)xf(x)\,dx.~~~~~~~(4.2)

Using the probability distribution function F(x), the mathematical expectation of a random variable can be expressed as follows:

M(X)=\int\limits_(-\infty)^(\infty)x\,d(F(x)).

Properties of mathematical expectation

Property 1. The mathematical expectation of the sum of two random variables is equal to the sum of their mathematical expectations:

M(X+Y)=M(X)+M(Y).

Property 2. The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations:

M(XY)=M(X)M(Y).

Property 3. The mathematical expectation of a constant value is equal to the constant itself:

M(c)=c.

Property 4. The constant multiplier of a random variable can be taken out of the sign of the mathematical expectation:

M(cX)=cM(X).

Property 5. The mathematical expectation of the deviation of a random variable from its mathematical expectation is equal to zero:

M(X-M(X))=0.

Example 3. Find the mathematical expectation of the number of defective products in a sample of five products, if the random variable X (the number of defective products) is given by a distribution series.

\begin(array)(|c|c|c|c|c|c|c|)\hline(X)&0&1&2&3&4&5\\\hline(P)&0,\!2373&0,\!3955&0,\!2637&0,\ !0879&0,\!0146&0,\!0010\\\hline\end(array)

Solution. Using formula (4.1) we find

M(X)=0\cdot0,\!2373+1\cdot0,\!3955+2\cdot0,\!2637+3\cdot0,\!0879+4\cdot0,\!0146+5\cdot0,\ !0010 =1,\!25.

Mode M_0 of a discrete random variable its most probable value is called.

Mode M_0 of a continuous random variable its value is called, which corresponds to the largest value of the distribution density. Geometrically, the mode is interpreted as the abscissa of the global maximum point of the distribution curve (Fig. 12).

Median M_e of a random variable its value is called for which the equality is true

P\(X M_e\).

From a geometric point of view, the median is the abscissa of the point at which the area of the figure, bounded by the probability distribution curve and the abscissa, is divided in half (Fig. 12). Since the entire area bounded by the distribution curve and the x-axis is equal to unity, the distribution function at the point corresponding to the median is equal to 0.5, i.e.

F(M_e)=P\(X

Using dispersion and standard deviation, one can judge the dispersion of a random variable around the mathematical expectation. As a measure of the dispersion of a random variable, the mathematical expectation of the squared deviation of the random variable from its mathematical expectation is used, which is called variance of a random variable X and denote D[X] :

D[X]=M((X-M(X))^2).

For a discrete random variable, the variance is equal to the sum of the products of the squared deviations of the values of the random variable from its mathematical expectation and the corresponding probabilities:

D[X]=\sum\limits_(i=1)^(n)(x_i-M(X))^2p_i.

For a continuous random variable whose distribution law is specified by the probability distribution density f(x) , the variance

D[X]=\int\limits_(-\infty)^(+\infty)(x-M(X))^2f(x)\,dx.

The dimension of variance is equal to the square of the dimension of the random variable and therefore cannot be interpreted geometrically. The standard deviation of a random variable, which is calculated by the formula, does not have these disadvantages

\sigma=\sqrt(D[X]).

Properties of dispersion of random variables

Property 1. The variance of the sum of two independent random variables is equal to the sum of the variances of these variables:

D=D[X]+D[Y].

Property 2. The variance of a random variable is equal to the difference between the mathematical expectation of the square of the random variable X and the square of its mathematical expectation:

D[X]=M(X^2)-(M(X))^2.~~~~~~~(4.3).

Property 3. The variance of a constant value is zero:

D[c]=0.

Property 4. The constant multiplier of a random variable can be taken out of the dispersion sign by first squaring it:

D=c^2D[X].

Property 5. The variance of the product of two independent random variables X and Y is determined by the formula

D=D[X]D[Y]+(M(X))^2D[Y]+(M(X))^2D[X].

Example 4. Calculate the variance of the number of defective products for the distribution of example 3.

Solution. By definition of variance

A generalization of the basic numerical characteristics of a random variable is the concept of moments of a random variable.

Initial moment of qth order a random variable is the mathematical expectation of the value X^q:

The initial moment of the first order represents the mathematical expectation, and the central moment of the second order represents the variance of the random variable.

The third-order normalized central moment characterizes the skewness or asymmetry of the distribution ( asymmetry coefficient):

A_s=\frac(\mu_(()_3))(\sigma^3).

The normalized central moment of the fourth order serves as a characteristic of the peakedness or flatness of the distribution ( excess):

E=\frac(\mu_(()_4))(\sigma^4)-3.

Example 5. Random variable X is specified by the probability distribution density

F(x)=\begin(cases)0,&x<0;\\ax^2,&02.\end(cases).

Find coefficient a, mathematical expectation, dispersion, skewness and kurtosis.

Solution. The area limited by the distribution curve is numerically equal to

\int\limits_(0)^(2)f(x)\,dx=a\int\limits_(0)^(2)x^2\,dx=\left.(a\,\frac(x^ 3)(3))\right|_(0)^(2)=\frac(8)(3)\,a.

Considering that this area should be equal to unity, we find a=\frac(3)(8) . Using formula (4.2) we find the mathematical expectation:

M(X)=\int\limits_(0)^(2)xf(x)\,dx=\frac(3)(8)\int\limits_(0)^(2)x^3\,dx= \left.(\frac(3)(8)\cdot\frac(x^4)(4))\right|_(0)^(2)=1,\!5.

Let us determine the dispersion using formula (4.3). To do this, we first find the mathematical expectation of the square of the random variable:

M(X^2)=\int\limits_(0)^(2)x^2f(x)\,dx=\frac(3)(8)\int\limits_(0)^(2)x^4 \,dx=\left.(\frac(3)(8)\cdot\frac(x^5)(5))\right|_(0)^(2)=2,\!4.

Thus,

\begin(aligned)D(X)&=M(X^2)-(M(X))^2=2,\!4-(1,\!5)^2=0,\!15;\ \ \sigma(X)&=\sqrt(D(X))=\sqrt(0,\!15)\approx0,\!3873.\end(aligned)

Using the initial moments, we calculate the central moments of the third and fourth order:

\begin(aligned)\nu_1&=M(X)=1,\!5;\quad\nu_2=M(X^2)=2,\!4.\\ \nu_3&=M(X^3)=\ int\limits_0^2(x^3f(x)\,dx)=\frac(3)(8)\int\limits_0^2(x^5\,dx)=\left.(\frac(3)( 8)\cdot\frac(x^6)(6))\right|_0^2=4;\\ \nu_4&=M(X^4)=\int\limits_0^2(x^4f(x)\ ,dx)=\frac(3)(8)\int\limits_0^2(x^6\,dx)=\left.(\frac(3)(8)\cdot\frac(x^7)(7 ))\right|_0^2\approx6,\!8571;\\ \mu_3&=\nu_3-3\nu_1\nu_2+2\nu_1^3=4-3\cdot1,\!5\cdot2,\!4 +2\cdot(1,\!5)^3=-0,\!05.\\ \mu_4&=\nu_4-4\nu_1\nu_3+6\nu_1^2\nu_2-3\nu_1^4=\ \&=6,\!8571-4\cdot1,\!5\cdot4+6\cdot(1,\!5)^2\cdot2,\!4-3\cdot(1,\!5)^4 =0,\!0696.\\ A_s&=\frac(\mu_3)(\sigma^3)=-\frac(0,\!05)((0,\!3873)^3)=-0,\ !86.\\ E&=\frac(\mu_4)(\sigma^4)-3=\frac(0,\!0696)((0,\!3873)^4)-3=-0,\! 093.\end(aligned)

Numerical characteristics of the arithmetic mean of n independent random variables

Let x_1,x_2,\ldots,x_n- values of the random variable X obtained in n independent tests. The mathematical expectation of a random variable is M(X) , and its variance is D[X] . These values can be considered as independent random variables X_1,X_2,\ldots,X_n with the same mathematical expectations and variances:

M(X_i)=M(X); \quad D=D[X],~~i=1,2,\ldots,n.

The arithmetic mean of these random variables

\overline(X)=\sum\limits_(i=1)^(n)\frac(X_i)(n).

Using the properties of mathematical expectation and dispersion of a random variable, we can write:

\begin(aligned)M(\overline(X))&=M\!\left(\frac(1)(n)\sum\limits_(i=1)^(n)X_i\right)=\frac( 1)(n)\sum\limits_(i=1)^(n)M(X_i)=M(X).~~~~~~~(4.4)\\ D[\overline(X)]&= D\!\left[\frac(1)(n)\sum\limits_(i=1)^(n)X_i\right]=\frac(1)(n^2)\sum\limits_(i=1 )^(n)D=\frac(D[X])(n).~~~~~~~(4.5)\end(aligned)

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One of the most important basic concepts of probability theory is the concept of a random variable.

A random variable is a quantity that, as a result of experiment, can take on one or another value, and it is not known in advance which one.

Examples of random variables:

1) number of hits with three shots;

2) the number of calls received at the telephone exchange per day;

3) hit rate with 10 shots.

In all three of these examples, the random variables can take on separate, isolated values that can be enumerated in advance.

So, in example 1) these values are:

in example 2):

in example 3)

0; 0,1; 0,2; …; 1,0.

Such random variables that take only discrete values that can be enumerated in advance are called discontinuous or discrete random variables.

There are other types of random variables, for example:

1) abscissa of the point of impact when fired;

2) error in weighing the body on analytical balances;

3) the speed of the aircraft at the moment it reaches a given altitude;

4) the weight of a randomly taken grain of wheat.

The possible values of such random variables are not separated from each other; they continuously fill a certain gap, which sometimes has clearly defined boundaries, and more often – vague, vague boundaries.

Such random variables, the possible values of which continuously fill a certain interval, are called continuous random variables.

The concept of a random variable plays a very important role in probability theory. If the “classical” theory of probability operated primarily with events, then the modern theory of probability prefers, wherever possible, to operate with random variables.

Let us give examples of methods of transition from events to random variables typical for probability theory.

An experiment is carried out as a result of which some event may or may not appear. Instead of an event, we can consider a random variable, which is equal to 1 if the event occurs and equal to 0 if the event does not occur. The random variable is obviously discontinuous; it has two possible values: 0 and 1. This random variable is called the characteristic random variable of the event. In practice, it often turns out to be more convenient to operate with their characteristic random variables instead of events. For example, if a series of experiments are carried out, in each of which the occurrence of the event is possible, then the total number of occurrences of the event is equal to the sum of the characteristic random variables of the event in all experiments. When solving many practical problems, using this technique turns out to be very convenient.

On the other hand, very often in order to calculate the probability of an event it turns out to be convenient to associate this event with some kind of continuous random variable (or system of continuous variables).

Let, for example, measure the coordinates of some object O in order to construct a point M, depicting this object in a panorama (scan) of the area. We are interested in the event that the error R in the position of point M will not exceed the specified value (Fig. 2.4.1). Let us denote random errors in measuring the coordinates of an object. Obviously, the event is equivalent to a random point M with coordinates falling within a circle of radius with a center at point O. In other words, for the event to occur, the random variables and must satisfy the inequality

The probability of an event is nothing more than the probability of inequality (2.4.1) being satisfied. This probability can be determined if the properties of random variables are known.

Such an organic connection between events and random variables is very characteristic of modern probability theory, which, wherever possible, moves from the “scheme of events” to the “scheme of random variables.” The latter scheme, compared to the first, is a much more flexible and universal apparatus for solving problems related to random phenomena.

Random variable- this is a quantity that, as a result of experiment, takes on one of many values, and the appearance of one or another value of this quantity cannot be accurately predicted before its measurement.

The formal mathematical definition is as follows: let be a probability space, then a random variable is a function measurable with respect to the Borel σ-algebra on . The probabilistic behavior of an individual (independent of others) random variable is completely described by its distribution.

Definition [edit]

Space of elementary events [edit]

The space of elementary events in the case of throwing a die

When a die is thrown, the top face may end up on one of six faces with a number of dots ranging from one to six. The loss of any edge in this case in probability theory is called an elementary event, that is

The set of all faces forms a space of elementary events, subsets of which are called random events. In the case of throwing a dice once, examples of events are

Algebra of events [edit]

A set of random events forms an event algebra if the following conditions are met:

If instead of the third condition it satisfies another condition: the union of a countable subfamily of also belongs to , then the set of random events forms a σ-algebra of events.

The algebra of events is a special case of the σ-algebra of sets.

The smallest among all possible -algebras, the elements of which are all intervals on the real line, is called Borel σ-algebra on the set of real numbers.

Probability [edit]

If each elementary event is associated with a number for which the condition is satisfied:

then it is considered that the probabilities of elementary events are given. The probability of an event, as a countable subset of the space of elementary events, is defined as the sum of the probabilities of those elementary events that belong to this event. The countability requirement is important because otherwise the amount will not be determined.

Let's consider an example of determining the probability of various random events. For example, if an event is an empty set, then its probability is zero:

If an event is a space of elementary events, then its probability is equal to one:

The probability of an event (a subset of the space of elementary events) is equal to the sum of the probabilities of those elementary events that the event in question includes.

Definition of a random variable [edit]

A random variable is a function that is measurable with respect to the Borel σ-algebra on .

A random variable can be defined in another equivalent way. A function is called a random variable if, for any real numbers and a set of events such that , belongs to .

Examples [edit]

equal to the arithmetic mean of all accepted values.

that is, the mathematical expectation is not defined.

Classification [edit]

Random variables can take discrete, continuous, and discrete-continuous values. Accordingly, random variables are classified into discrete, continuous and discrete-continuous (mixed).

In the test scheme, both a separate random variable (one-dimensional/scalar) and a whole system of one-dimensional interrelated random variables (multidimensional/vector) can be defined.

An example of a mixed random variable is the waiting time when crossing a road in a city at an uncontrolled intersection.
In infinite schemes (discrete or continuous), it is convenient to describe the initially elementary outcomes quantitatively. For example, numbers of gradations of types of accidents when analyzing road accidents; uptime of the device during quality control, etc.
Numerical values describing the results of experiments may not necessarily characterize individual elementary outcomes in the test scheme, but also correspond to some more complex events.

On the one hand, several numerical values that need to be analyzed together can be simultaneously associated with one test scheme and with individual events in it.

For example, the coordinates (abscissa, ordinate) of some kind of shell explosion when firing at a ground target; metric dimensions (length, width, etc.) of parts during quality control; results of a medical examination (temperature, pressure, pulse, etc.) when diagnosing the patient; Population census data (by age, gender, income, etc.).

Since the values of the numerical characteristics of the test schemes correspond to some random events in the scheme (with their certain probabilities), then these values themselves are random (with the same probabilities). Therefore, such numerical characteristics are usually called random variables. In this case, the distribution of probabilities according to the values of a random variable is called the law of distribution of a random variable.

Description methods [edit]

You can partially specify a random variable, thereby describing all its probabilistic properties as a separate random variable, using the distribution function, probability density and characteristic function, determining the probabilities of its possible values. The distribution function F(x) is the probability that the values of the random variable are less than the real number x. From this definition it follows that the probability of a random variable falling into the interval

A random variable, generally speaking, can take values in any measurable space. Then it is more often called a random vector or a random element. For example,

See also [edit]

Random process
Distribution function
Expectation

Notes [edit]

1 2 Chernova N. I. Chapter 1. § 2. Elementary theory of probability // Theory of Probability. - Study guide. - Novosibirsk: Novosibirsk State. univ., 2007. - 160 p.
Chernova N. I. Chapter 3. § 1. Algebra and sigma-algebra of events // Theory of Probability. - Study guide. - Novosibirsk: Novosibirsk State. univ., 2007. - 160 p.
Chernova N. I. CHAPTER 1 § 2. Elementary theory of probability // Theory of probability. - Study guide. - Novosibirsk: Novosibirsk State. univ., 2007. - 160 p.
1 2 Chernova N. I. Chapter 6. Random variables and their distributions § 1. Random variables // Theory of Probability. - Study guide. - Novosibirsk: Novosibirsk State. univ., 2007. - 160 p.

Literature [edit]

Gnedenko B.V. Probability theory course. - 8th ed. add. and corr. - M.: Editorial URSS, 2005. - 448 p.
Mathematical encyclopedic dictionary / Ch. ed. Prokhorov Yu. V.. - 2nd ed. - M.: “Soviet Encyclopedia”, 1998. - 847 p.
Tikhonov V.I., Kharisov V.N. Statistical analysis and synthesis of radio engineering devices and systems. - Textbook for universities. - M.: Radio and Communications, 1991. - 608 p. - ISBN 5-256-00789-0
Chernova N. I. Probability theory. - Study guide. - Novosibirsk: Novosibirsk State. univ., 2007. - 160 p.

Definition of a random variable. Many random events can be quantified by random variables.

Random is a quantity that takes values depending on a combination of random circumstances.

Random variables are: the number of patients at a doctor's appointment, the number of students in the audience, the number of births in a city, the life expectancy of an individual, the speed of a molecule, air temperature, the error in measuring a value, etc. If you number the balls in an urn something like this, as is done when playing a lotto draw, then randomly removing a ball from the urn will show a number that is a random variable.

There are discrete and continuous random variables.

A random variable is called discrete if it takes a countable set of values: the number of letters on an arbitrary page of a book, the energy of an electron in an atom, the number of hairs on a person’s head, the number of grains in ears of corn, the number of molecules in a given volume of gas, etc.

A continuous random variable takes any values within a certain interval: body temperature, grain weight V ears of wheat, the coordinate of the location where the bullet hits the target (we take the bullet as a material point), etc.

Distribution of a discrete random variable. A discrete random variable is considered given if its possible values and the corresponding probabilities are indicated. Let us denote a discrete random variable X, its meanings x 1 x 2,…., and the probabilities P(x 1)= p 1, P(x 2)= p 2 etc. Collection X And P is called the distribution of a discrete random variable(Table 1).

Table 1

The random variable is the number of the sport in the game “Sportlo-10”. The total number of species is 49. Indicate the distribution of this random variable (Table 3).

Table 3

Meaning 1 = 0 corresponds to the case in which the event three times in a row A didn't happen. The probability of this complex event, according to the probability multiplication theorem (2.6), is equal to

Meaning I = 1 refers to the case in which event A occurred in one of the three trials. Using formula (2.6) we obtain

Since when l = 1 two other complex events also occur: (A and A and A) and (A and A and A), then it is necessary, using the probability addition theorem (2.4), to obtain the total probability for l = 1, adding the previous expression three times:

Meaning I = 2 corresponds to the case in which event A occurred in two of the three trials. Using arguments similar to those given above, we obtain the total probability for this case:

At 1 = 3 event A appears in all three trials. Using the probability multiplication theorem, we find

In general, the binomial distribution allows us to determine the probability that event A will occur l times at n tests:

Based on many years of observations, calling a doctor to a given house is estimated with a probability of 0.5. Find the probability that four doctor calls will occur within six days; P(A)= 0,5, n = 6,1 = 4. T Let's use formula (2.10):

Numerical characteristics of a discrete random variable. In many cases, along with the distribution of a random variable or instead of it, information about these quantities can be provided by numerical parameters called numerical characteristics of a random variable. Let's look at the most common of them.

The mathematical expectation (average value) of a random variable is the sum of the products of all its possible values
on the probabilities of these values:

Let, with a large number of tests n discrete random variable X takes values x v x 2,..., x n respectively m 1, m g,..., t p once. The average value is

If n is large, then the relative frequencies t 1 / p, t 2 / p,... will tend to the probabilities, and the average value will tend to the mathematical expectation. That is why the mathematical expectation is often identified with the average value.

Find the mathematical expectation for a discrete random variable, which is specified by the number on the face when throwing a dice (see Table 2).

We use formula (2.11):

Find the mathematical expectation for a discrete random variable, which is determined by the Sportloto circulation (see Table 3). According to formula (2.11), we find

Possible values of a discrete random variable are scattered around its mathematical expectation, some of them exceed M(X), part - less M(X). How to estimate the degree of dispersion of a random variable relative to its mean value? It may seem that to solve such a problem one should calculate the deviations of all random variables from its mathematical expectation X - M(X), and then find the mathematical expectation (average value) of these deviations: M[X - M(X)]. Taking the proof, we note that this value is equal to zero, since the deviations of random variables from the mathematical expectation have both positive and negative values. Therefore, it is advisable to take into account either the absolute values of deviations M[X - M(X)], or their squares M[X - M(X)] 2 . The second option turns out to be preferable, and this is how we come to the concept of the dispersion of a random variable.

The variance of a random variable is the mathematical expectation of the squared deviation of a random variable from its mathematical expectation:

It means that the variance is equal to the difference between the mathematical expectation of the square of the random variable X and the square of its mathematical expectation.

Find the variance of the random variable, which is given by the number on the edge when throwing a die (see Table 2).

The mathematical expectation of this distribution is 3.5. Let us write down the values of the squares of the deviation of random variables from the mathematical expectation: (1 - 3.5) 2 = 6.25; (2 - 3.5) 2 = 2.25; (3 - 3.5) 2 = 0.25; (4 - 3.5) 2 = 0.25; (5 - 3.5) 2 = 2.25; (6 - 3.5) 2 = 6.25. Using formula (2.12), taking into account (2.11), we find the variance:

As follows from (2.12), the variance has the dimension of the square of the dimension of the random variable. In order to estimate the distance of a random variable in units of the same dimension, the concept is introduced standard deviation, which is understood as the square root of the variance:

Distribution and characteristics of a continuous random variable. A continuous random variable cannot be specified by the same distribution law as a discrete one. In this case, proceed as follows.

Let dP be the probability that a continuous random variable X takes values between X And X+ dx. Obviously, Irm is a longer interval dx, the greater the probability dP: dP ~ dx. Moreover, the probability must also depend on the random Quantity itself, near which the interval is located, therefore

Where f(x)- probability density, or probability distribution function. It shows how the probability related to the interval changes dx random variable, depending on the value of this variable itself:

Integrating expression (2.15) within the appropriate limits, we find the probability that the random variable takes any value in the interval (ab):

The normalization condition for a continuous random variable has the form

As can be seen from (2.19), this function is equal to the probability that the random variable takes values less than X:

For a continuous random variable, the mathematical expectation and variance are written respectively in the form