Based on the geometric meaning of the integral as an area, we can say that the probability of fulfilling the inequalities is equal to the area of ​​a curvilinear trapezoid with base bounded above by a curve (Fig. 6).

Since, and on the basis of formula (22)

Using formula (22), we find as the derivative of the integral with respect to the variable upper boundary, assuming the distribution density to be continuous**:

Note that for a continuous random variable, the distribution function F(x) continuous at any point X, where the function is continuous. This follows from the fact that F(x) is differentiable at these points.

Based on formula (23), assuming x 1 =x, we have

Due to the continuity of the function F(x) we get that

Hence

Thus, probability Togo, What continuous random magnitude Maybe accept any separate meaning X, is equal to zero.

It follows from this that the events consisting in the fulfillment of each of the inequalities

They have the same probability, i.e.

Indeed, for example,

Comment. As we know, if an event is impossible, then the probability of its occurrence is zero. In the classical definition of probability, when the number of test outcomes is finite, the reverse proposition also takes place: if the probability of an event is zero, then the event is impossible, since in this case none of the test outcomes favors it. In the case of a continuous random variable, the number of its possible values ​​is infinite. The probability that this value will take on any particular value x 1 as we have seen, is equal to zero. However, it does not follow from this that this event is impossible, since as a result of the test, the random variable can, in particular, take on the value x 1 . Therefore, in the case of a continuous random variable, it makes sense to talk about the probability that the random variable falls into the interval, and not about the probability that it will take on a particular value.

So, for example, in the manufacture of a roller, we are not interested in the probability that its diameter will be equal to the nominal value. For us, the probability that the diameter of the roller does not go out of tolerance is important.

Example. The distribution density of a continuous random variable is given as follows:

The graph of the function is shown in Fig. 7. Determine the probability that a random variable will take a value that satisfies the inequalities. Find the distribution function of a given random variable. ( Solution)

The next two paragraphs are devoted to the distributions of continuous random variables that are often encountered in practice - uniform and normal distributions.

* A function is called piecewise continuous on the entire numerical axis if it is either continuous on any segment or has a finite number of discontinuity points of the first kind.

** The rule for differentiating an integral with a variable upper bound, derived in the case of a finite lower bound, remains valid for integrals with an infinite lower bound. Indeed,

Since the integral

is a constant value.

random variables

Under random variables understand the numerical characteristics of random events. In other words, random variables are the numerical results of experiments, the values ​​of which cannot (at a given time) be predicted in advance.

For example, the following quantities can be considered as random:

2. The percentage of boys among children born in a given maternity hospital on some specific day.

3. The number and area of ​​sunspots visible at some observatory during a given day.

4. The number of students who were late for this lecture.

5. The dollar exchange rate on the stock exchange (say, on the MICEX), although it may not be so “random”, as it seems to the inhabitants.

6. The number of equipment failures on a given day at a particular enterprise.

Random variables are divided into discrete and continuous depending on whether the set of all possible values ​​of the corresponding characteristic is discrete or continuous.

This division is rather conditional, but it is useful in choosing adequate research methods. If the number of possible values ​​of a random variable is finite or comparable to the set of all natural numbers (ie, can be renumbered), then the random variable PDF created with FinePrint pdfFactory trial version http://www.fineprint.com is called discrete. Otherwise, it is called continuous, although in fact, as if implicitly, it is assumed that actually continuous random variables take their values ​​in some simple numerical interval (segment, interval). For example, random variables will be discrete, given above under numbers 4 and 6, and continuous - under numbers 1 and 3 (spot areas). Sometimes the random variable has a mixed character. Such, for example, is the exchange rate of the dollar (or some other currency), which in fact takes only a discrete set of values, but it turns out to be convenient to consider that the set of its values ​​is “continuous”.

Random variables can be specified in different ways.

Discrete random variables are usually given by their own distribution law. Here, each possible value x1, x2,... of the random variable X is associated with the probability p1,p2,... of this value. The result is a table consisting of two rows:

This is the law of distribution of a random variable.

It is impossible to specify continuous random variables by distribution laws, since, by their very definition, their values ​​cannot be renumbered, and therefore the assignment in the form of a table is excluded here. However, for continuous random variables there is another way of specifying (applicable, by the way, for discrete variables) - this is the distribution function:

equal to the probability of the event , which consists in the fact that the random variable X takes a value less than a given number x.

Often, instead of the distribution function, it is convenient to use another function - the distribution density f(x) of the distribution of a random variable X. It is also sometimes called the differential distribution function, and F(x) in this terminology is called the integral distribution function. These two functions mutually determine each other by the following formulas:

If the random variable is discrete, then the concept of the distribution function also makes sense for it, in this case the distribution function graph consists of horizontal sections, each of which is located above the previous one by an amount equal to pi.

Important examples of discrete quantities are, for example, binomially distributed quantities (Bernoulli distribution), for which PDF created with FinePrint pdfFactory trial version http://www.fineprint.com

pk(1-p)n-k= !()!

where p is the probability of a single event (it is sometimes conditionally called the “probability of success”). This is how the results of a series of successive homogeneous tests are distributed (Bernoulli scheme). The limiting case of the binomial distribution (as the number of trials increases) is the Poisson distribution, for which

pk=?k/k! exp(-?),

where?>0 is some positive parameter.

The simplest example of a continuous distribution is the uniform distribution. It has a constant distribution density on the segment, equal to 1 / (b-a), and outside this segment, the density is 0.

An extremely important example of a continuous distribution is the normal distribution. It is given by two parameters m and? (expectation and standard deviation - see below), its distribution density has the form:

1 exp(-(x-m)2/2?2)

The fundamental role of the normal distribution in probability theory is explained by the fact that, by virtue of the Central Limit Theorem (CLT), the sum of a large number of random variables that are pairwise independent (see below about the concept of independence of random variables) or weakly dependent turns out to be approximately distributed according to the normal law. Hence it follows that a random variable, the randomness of which is caused by the superposition of a large number of random factors that are weakly dependent on each other, can be considered approximately as normally distributed (regardless of how the factors constituting it were distributed). In other words, the normal distribution law is very universal.

There are several numerical characteristics that are convenient to use when studying random variables. Among them, we single out the mathematical expectation

equal to the mean value of the random variable, the variance

D(X)=M(X-M(X))2,

equal to the mathematical expectation of the square of the deviation of the random variable from the mean value, and one more additional value convenient in practice (of the same dimension as the original random variable):

called the standard deviation. We will assume (without stipulating this further) that all the written integrals exist (i.e., converge on the entire real axis). As is known, the variance and standard deviation characterize the degree of dispersion of a random variable around its mean value. The smaller the dispersion, the more closely the values ​​of a random variable cluster around its mean value.

For example, the mean for a Poisson distribution is ?, for a uniform distribution it is (a+b)/2, and for a normal distribution it is m. The variance for the Poisson distribution is ?, for the uniform distribution (b-a)2/12, and for the normal distribution is ?2. In what follows, the following properties of mathematical expectation and variance will be used:

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

3. D(cX)=c2D(X), where c is an arbitrary constant number.

4. D(X+A)=D(A) for an arbitrary constant (non-random) value A.

The random variable?=U-MU is called centered. It follows from property 1 that M?=M(U-MU)=M(U)-M(U)=0, that is, its average value is 0 (here is its name). Moreover, due to property 4, we have D(?)=D(U).

There is also a useful relation that is convenient to use in practice to calculate the variance and related quantities:

5. D(X)=M(X2)-M(X)2

Random variables X and Y are called independent if, for their arbitrary values ​​x and y, respectively, the events and are independent. For example, the results of measuring the voltage in the power grid and the growth of the main power engineer of the enterprise will be independent (apparently ...). But the capacity of this power grid and the salary of the chief power engineer at enterprises can no longer always be considered independent.

If the random variables X and Y are independent, then the following properties also hold (which may not hold for arbitrary random variables):

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

6. D(X+Y)=D(X)+D(Y).

In addition to individual random variables X,Y,..., systems of random variables are also studied. For example, a pair of (X,Y) random variables can be considered as a new random variable whose values ​​are two-dimensional vectors. Similarly, systems of a larger number of random variables, called multidimensional random variables, can be considered. Such systems of quantities are also given by their distribution function. For example, for a system of two random variables, this function has the form

F(x,y)=P,

that is, it is equal to the probability of the event that the random variable X takes a value less than a given number x, and the random variable Y is less than a given number y. This function is also called the joint distribution function of random variables X and Y. You can also consider the average vector - a natural analogue of the mathematical expectation, but instead of the variance, you have to study several numerical characteristics, called second-order moments. These are, firstly, two partial variances DX and DY PDF created with FinePrint pdfFactory trial version http://www.fineprint.com of random variables X and Y, considered separately, and, secondly, the covariance moment, in more detail discussed below.

If random variables X and Y are independent, then

F(x,y)=FX(x)FY(y)

The product of the distribution functions of random variables X and Y, and therefore the study of a pair of independent random variables, is reduced in many respects simply to the study of X and Y separately.

random variables

Experiments were considered above, the results of which are random events. However, it often becomes necessary to quantitatively represent the results of an experiment in the form of a certain quantity, which is called a random variable. A random variable is the second (after a random event) main object of study of probability theory and provides a more general way of describing an experience with a random outcome than a collection of random events.

Considering experiments with a random outcome, we have already dealt with random variables. So, the number of successes in a series of trials is an example of a random variable. Other examples of random variables are: the number of calls at the telephone exchange per unit of time; waiting time for the next call; the number of particles with a given energy in systems of particles considered in statistical physics; average daily temperature in a given area, etc.

A random variable is characterized by the fact that it is impossible to accurately predict its value, which it will take, but on the other hand, the set of its possible values ​​\u200b\u200bis usually known. So for the number of successes in a sequence of trials, this set is finite, since the number of successes can take on values. The set of values ​​of a random variable can coincide with the real semi-axis, as in the case of waiting time, etc.

Let us consider examples of experiments with a random outcome, which are usually described by random events, and introduce an equivalent description by specifying a random variable.

1). Let the result of an experience be an event or an event. Then this experiment can be associated with a random variable that takes two values, for example, and with probabilities and, moreover, the equalities take place: and. Thus, an experience is characterized by two outcomes with probabilities and, or the same experience is characterized by a random variable that takes two values ​​and with probabilities and.

2). Consider the experiment with throwing a dice. Here, the outcome of the experiment can be one of the events, where is the loss of a face with a number. probabilities. Let us introduce an equivalent description of this experiment using a random variable that can take values ​​with probabilities.

3). The sequence of independent tests is characterized by a complete group of incompatible events, where is an event consisting in the appearance of success in a series of experiments; moreover, the probability of an event is determined by the Bernoulli formula, i.e. Here you can enter a random variable - the number of successes, which takes values ​​with probabilities. Thus, a sequence of independent trials is characterized by random events with their probabilities or by a random variable with probabilities that it takes values.

4). However, not for any experience with a random outcome there is such a simple correspondence between a random variable and a set of random events. For example, consider an experiment in which a point is randomly thrown onto a line. Here it is natural to introduce a random variable - the coordinate on the segment in which the point falls. Thus, we can talk about a random event, where is the number of. However, the probability of this event. You can do otherwise - divide the segment into a finite number of non-intersecting segments and consider random events, consisting in the fact that the random variable takes values ​​from the interval. Then the probabilities are finite. However, this method also has a significant drawback, since the segments are chosen arbitrarily. In order to eliminate this shortcoming, segments of the form where the variable is considered. Then the corresponding probability is a function of the argument. This complicates the mathematical description of the random variable, but at the same time the description (29.1) becomes the only one, and the ambiguity of the choice of segments is eliminated.

For each of the examples considered, it is easy to determine the probability space, where is the space of elementary events, is the algebra of events (subsets), is the probability defined for any. For example, in the last example, - is the algebra of all segments contained in.

The considered examples lead to the following definition of a random variable.

Let be a probability space. A random variable is a single-valued real function defined on, for which the set of elementary events of the form is an event (i.e. belongs) for each real number.

Thus, the definition requires that for each real set, and this condition ensures that the probability of an event is defined for each. This event is usually denoted by a shorter record.

Probability Distribution Function

The function is called the probability distribution function of the random variable.

The function is sometimes called briefly - the distribution function, and also - the integral law of the probability distribution of a random variable. A function is a complete characteristic of a random variable, that is, it is a mathematical description of all properties of a random variable, and there is no more detailed way to describe these properties.

We note the following important feature of the definition (30.1). Often a function is defined differently:

According to (30.1), the function is right-continuous. This issue will be considered in more detail below. If, however, definition (30.2) is used, then - is continuous on the left, which is a consequence of the application of strict inequality in relation (30.2). Functions (30.1) and (30.2) are equivalent descriptions of a random variable, since it does not matter which definition to use both when studying theoretical issues and when solving problems. For definiteness, in what follows we will use only definition (30.1).

Consider an example of plotting a function graph. Let a random variable take values, with probabilities, moreover. Thus, this random variable takes other values ​​except those indicated with zero probability:, for any,. Or, as they say, a random variable cannot take on other values. Let for definiteness. Find the values ​​of the function for from the intervals: 1), 2), 3), 4), 5), 6), 7). On the first interval, so the distribution function. 2). If, then. Obviously random events and are incompatible, therefore, according to the formula for adding probabilities. By condition, the event is impossible and, a. That's why. 3). Let, then. Here the first term, and the second, because the event is impossible. Thus for anyone satisfying the condition. 4). Let, then. 5). If, then. 6) When we have. 7) If, then. The calculation results are shown in Figs. 30.1 function graph. At the discontinuity points, the continuity of the function on the right is indicated.

Basic properties of the probability distribution function

Consider the main properties of the distribution function, which follow directly from the definition:

1. Let's introduce the notation:. Then it follows from the definition. Here, the expression is treated as an impossible event with zero probability.

2. Let. Then it follows from the definition of the function. A random event is certain and its probability is equal to one.

3. The probability of a random event, consisting in the fact that a random variable takes a value from the interval at is determined through a function by the following equality

To prove this equality, consider the relation.

The events and are inconsistent, therefore, according to the formula for adding probabilities, it follows from (31.3) that and coincides with formula (31.2), since and.

4. The function is non-decreasing. Let's look at the proof. In this case, equality (31.2) is valid. Its left side, since the probability takes values ​​from the interval. Therefore, the right side of equality (31.2) is also non-negative:, or. This equality is obtained under the condition, therefore, is a non-decreasing function.

5. The function is right continuous at every point, i.e.

where is any sequence tending to the right, i.e. And.

To prove it, we represent the function in the form:

Now, based on the axiom of countable additivity of probability, the expression in curly brackets is equal, thus, which proves the right continuity of the function.

Thus, each probability distribution function has properties 1-5. The converse statement is also true: if, satisfies conditions 1-5, then it can be considered as a distribution function of some random variable.

Probability distribution function of a discrete random variable

A random variable is called discrete if the set of its values ​​is finite or countable.

For a complete probabilistic description of a discrete random variable that takes on values, it suffices to specify the probabilities that the random variable takes on a value. If and are given, then the probability distribution function of a discrete random variable can be represented as:

Here the summation is carried out over all indices that satisfy the condition.

The probability distribution function of a discrete random variable is sometimes represented in terms of the so-called unit jump function.

In this case, it takes the form if the random variable takes on a finite set of values, and the upper summation limit in (32.4) is assumed to be equal if the random variable takes on a countable set of values.

An example of constructing a graph of the probability distribution functions of a discrete random variable was considered in Section 30.

Probability density

Let a random variable have a differentiable probability distribution function, then the function is called the probability distribution density (or probability density) of the random variable, and the random variable is called a continuous random variable.

Consider the basic properties of the probability density.

The definition of the derivative implies the equality:

According to the properties of the function, equality takes place. Therefore (33.2) takes the form:

This relation explains the name of the function. Indeed, according to (33.3), the function is the probability per unit interval, at the point, since. Thus, the probability density defined by relation (33.3) is similar to the definitions of the densities of other quantities known in physics, such as current density, matter density, charge density, etc.

2. Since is a non-decreasing function, then its derivative is a non-negative function:

3. It follows from (33.1) because. Thus, the equality

4. Since, it follows from relation (33.5)

Equality, which is called the normalization condition. Its left side is the probability of a certain event.

5. Let, then from (33.1) it follows

This relationship is important for applications because it allows you to calculate the probability in terms of the probability density or in terms of the probability distribution function. If we set, then relation (33.6) follows from (33.7).

On fig. 33.1 shows examples of graphs of the distribution function and probability density.

Note that the probability distribution density can have several maxima. The value of the argument at which the density has a maximum is called the distribution mode of the random variable. If the density has more than one mode, then it is called multimodal.

Probability density of a discrete random variable

distribution discrete probability density

Let a random variable take values ​​with probabilities,. Then its probability distribution function is where is the unit jump function. It is possible to determine the probability density of a random variable by its distribution function, taking into account equality. However, mathematical difficulties arise in this case, due to the fact that the unit jump function in (34.1) has a discontinuity of the first kind at. Therefore, the derivative of the function does not exist at the point.

To overcome this complexity, a -function is introduced. The unit jump function can be represented in terms of the -function by the following equality:

Then formally the derivative and probability density of a discrete random variable is determined from relation (34.1) as a derivative of the function:

The function (34.4) has all the properties of the probability density. Consider an example. Let a discrete random variable take values ​​with probabilities, and let . Then the probability - that the random variable will take a value from the segment can be calculated based on the general properties of the density according to the formula:

Here, since the singular point of the function determined by the condition is inside the integration region at, and at the singular point is outside the integration region. Thus.

The function (34.4) also satisfies the normalization condition:

Note that in mathematics, a record of the form (34.4) is considered incorrect (incorrect), and the record (34.2) is considered correct. This is due to the fact that the -function with a zero argument, and say that it does not exist. On the other hand, in (34.2) the -function is contained under the integral. In this case, the right side of (34.2) is a finite value for any, i.e. the integral of the -function exists. Despite this, in physics, engineering and other applications of probability theory, the representation of density in the form (34.4) is often used, which, firstly, allows obtaining correct results by applying properties - functions, and secondly, has an obvious physical interpretation.

Examples of Densities and Probability Distributions

35.1. A random variable is called uniformly distributed on a segment if its probability distribution density

where is a number determined from the normalization condition:

Substitution (35.1) in (35.2) leads to equality, the solution of which relatively has the form:.

The probability distribution function of a uniformly distributed random variable can be found by formula (33.5), which determines through the density:

On fig. 35.1 graphs of functions and a uniformly distributed random variable are presented.

35.2. A random variable is called normal (or Gaussian) if its probability distribution density is:

where, are numbers called function parameters. When the function takes its maximum value:. The parameter has the meaning of effective width. In addition to this geometric interpretation, the parameters also have a probabilistic interpretation, which will be discussed later.

From (35.4) follows the expression for the probability distribution function

where is the Laplace function. On fig. 35.2 graphs of functions and a normal random variable are presented. To indicate that a random variable has a normal distribution with parameters and is often used notation.

35.3. A random variable has a Cauchy probability density if

This density corresponds to the distribution function

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

Let us define its probability distribution function. For from (35.8) it follows. If then

35.5. The Rayleigh probability distribution of a random variable is determined by the density of the form

This density corresponds to the probability distribution function at and equal to at.

35.6. Consider examples of constructing the distribution function and density of a discrete random variable. Let the random variable be the number of successes in a sequence of independent trials. Then the random variable takes values, with a probability, which is determined by the Bernoulli formula:

where, are the probabilities of success and failure in one experiment. Thus, the probability distribution function of a random variable has the form

where is the unit jump function. Hence the distribution density:

where is the delta function.

Singular random variables

In addition to discrete and continuous random variables, there are also so-called singular random variables. These random variables are characterized by the fact that their probability distribution function is continuous, but the growth points form a set of zero measure. The growth point of a function is the value of its argument such that the derivative.

Thus, almost everywhere on the domain of the function. A function that satisfies this condition is also called singular. An example of a singular distribution function is the Cantor curve (Fig. 36.1), which is constructed as follows. Relies on and on. Then the interval is divided into three equal parts (segments) and the value for the inner segment is determined - as a half-sum of the already determined values ​​on the nearest segments on the right and left. At the moment, a function is defined for, its value, and for with a value. The half-sum of these values ​​is equal to and determines the value on the inner segment. Then the segments and are considered, each of them is divided into three equal segments and the function is defined on the internal segments as a half-sum of the given values ​​of the function closest to the right and left. Thus, for a function - as a half-sum of numbers and. Similarly on the interval function. Then the function is defined on the interval, on which, etc.

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Let a continuous random variable X be given by the distribution function F(X) . Let us assume that all possible values ​​of the random variable belong to the segment [ A, B].

Definition. mathematical expectation continuous random variable X, the possible values ​​of which belong to the segment , is called a definite integral

If the possible values ​​of a random variable are considered on the entire number axis, then the mathematical expectation is found by the formula:

In this case, of course, it is assumed that the improper integral converges.

Definition. dispersion continuous random variable is called the mathematical expectation of the square of its deviation.

By analogy with the variance of a discrete random variable, the following formula is used for the practical calculation of the variance:

Definition. Standard deviation It is called the square root of the variance.

Definition. Fashion M0 of a discrete random variable is called its most probable value. For a continuous random variable, the mode is the value of the random variable at which the distribution density has a maximum.

If the distribution polygon for a discrete random variable or the distribution curve for a continuous random variable has two or more maxima, then such a distribution is called Dual modal or Multimodal.

If a distribution has a minimum but no maximum, then it is called Antimodal.

Definition. median The MD of a random variable X is its value, relative to which it is equally likely to obtain a larger or smaller value of the random variable.

Geometrically, the median is the abscissa of the point at which the area bounded by the distribution curve is divided in half.

Note that if the distribution is unimodal, then the mode and median coincide with the mathematical expectation.

Definition. Starting moment Order K A random variable X is the mathematical expectation of the value X K.

For a discrete random variable: .

.

The initial moment of the first order is equal to the mathematical expectation.

Definition. Central moment Order K random variable X is called the mathematical expectation of the value

For a discrete random variable: .

For a continuous random variable: .

The first order central moment is always zero, and the second order central moment is equal to the dispersion. The central moment of the third order characterizes the asymmetry of the distribution.

Definition. The ratio of the central moment of the third order to the standard deviation in the third degree is called Asymmetry coefficient.

Definition. To characterize the sharpness and flatness of the distribution, a quantity called kurtosis.

In addition to the quantities considered, the so-called absolute moments are also used:

Absolute starting moment: .

Absolute central moment: .

The absolute central moment of the first order is called Arithmetic mean deviation.

Example. For the example considered above, determine the mathematical expectation and variance of the random variable X.

Example. An urn contains 6 white and 4 black balls. A ball is removed from it five times in a row, and each time the ball taken out is returned back and the balls are mixed. Taking the number of extracted white balls as a random variable X, draw up the law of distribution of this quantity, determine its mathematical expectation and variance.

Since the balls in each experiment are returned back and mixed, the trials can be considered independent (the result of the previous experiment does not affect the probability of the occurrence or non-occurrence of an event in another experiment).

Thus, the probability of a white ball appearing in each experiment is constant and equal to

Thus, as a result of five successive trials, the white ball may not appear at all, appear once, twice, three, four or five times.

To draw up a distribution law, you need to find the probabilities of each of these events.

1) The white ball did not appear at all:

2) The white ball appeared once:

3) The white ball will appear twice: .

4) The white ball will appear three times:

random variables.

In mathematics magnitude is a common name for various quantitative characteristics of objects and phenomena. Length, area, temperature, pressure, etc. are examples of various quantities.

A value that takes various numerical values ​​under the influence of random circumstances, is called random variable. Examples of random variables: 1) the number of patients waiting to see a doctor, 2) the exact dimensions of the internal organs of people, etc.

There are discrete and continuous random variables.

The random variable is called discrete if it only accepts certain distinct values ​​that can be set and enumerated.

Examples:

1) the number of students in the audience - can only be a positive integer:

0,1,2,3,4….. 20…..

2) the number that appears on the top face when throwing a dice can only take integer values ​​from 1 to 6.

3) the relative frequency of hitting the target with 10 shots - its values:

0; 0,1; 0,2; 0,3 ….. 1

4) the number of events occurring at the same time intervals: pulse rate, the number of ambulance calls per hour, the number of operations per month with a fatal outcome, etc.

The random variable is called continuous if she can accept any values ​​within a certain interval, which sometimes has sharply defined boundaries, and if they are not known, then it is considered that the values ​​of the random variable X lie in the interval (-¥; ¥). Continuous random variables include, for example, temperature, pressure, weight and the height of people, the size of the blood cells, the pH of the blood, etc.

The concept of a random variable plays a decisive role in modern probability theory, which has developed special techniques for the transition from random events to random variables.

If a random variable depends on time, then we can talk about a random process.

3.1. Distribution law of a discrete random variable

To give a complete description of a discrete random variable, it is necessary to indicate all its possible values ​​and their probabilities.

The correspondence between the possible values ​​of a discrete random variable and their probabilities is called distribution law of this quantity.

Let us denote the possible values ​​of the random variable Х as хi, and the corresponding probabilities as рi* . Then the law of distribution of a discrete random variable can be specified in three ways: in the form of a table, graph or formula.

1. Table, which is called near distribution, all possible values ​​of a discrete random variable X and the probabilities P(X) corresponding to these values ​​are listed:

Table 3.1.

X

In this case, the sum of all probabilities pi should be equal to one ( normalization condition):

pi = p1 + p2 +...+pn=

2. Graphically- in the form of a broken line, which is commonly called distribution polygon(fig.3.1). Here, all possible values ​​of the random variable Xi are plotted along the horizontal axis, and the corresponding probabilities pi are plotted along the vertical axis.

3. Analytically- in the form of a formula: For example, if the probability of hitting the target with one shot is equal to R, then the probability of a miss with one shot q \u003d 1 - p, a. the probability of hitting the target 1 time at n shots is given by the formula: P(n) = qn-1×p,

3.2. The law of distribution of a continuous random variable. Probability distribution density.

For continuous random variables, it is impossible to apply the distribution law in the forms given above, since a continuous variable has an uncountable (“uncountable”) set of possible values ​​that completely fill a certain interval. Therefore, it is impossible to make a table in which all its possible values ​​\u200b\u200bare listed, or to build a distribution polygon. In addition, the probability of any particular value is very small (close to 0). At the same time, different areas (intervals) of possible values ​​of a continuous random variable are usually not equally probable. Thus, here too there is a certain distribution law, although not in the former sense.

Let us consider a continuous random variable X, the possible values ​​of which completely fill a certain interval (a, b)*. Law probability distributions such a value should allow us to find the probability of its value falling into any given interval (x1, x2) lying inside (a, b *) (Fig. 3.2.)

This probability is denoted by P(x1<Х< х2), или Р(х1 £ Х £ х2).

Consider first very short interval values ​​from x to (x + Dx) (see Fig. 3.2.) The small probability dР that the random variable X takes some value from this small interval (x, x + Dx) will be proportional to the value of this interval Dx: dР ~ Dx, or, by introducing the proportionality factor f, which itself may depend on x, we obtain:

dР = f(х) × Dх. (3.2)

The function we introduced here f(x) called probability density random variable X or, in short, the probability density (distribution density). Equation (3.2) can be considered as a differential equation, and then the probability of hitting was given. the rank of X in the interval (x1, x2) is equal to:

P (x1< Х < х2) = f(x) dx. (3.3)

Graphically, this probability P (x1< Х < х2) равна площади криволинейной трапеции, ограниченной осью абсцисс, кривой f(x) and straight lines X = x1 and X = x2 (see Fig.3.3), which follows from the geometric meaning of the definite integral (3.3). Curve f(x) it is called distribution curve.

It can be seen from (3.3) that if the function f(x), then by changing the limits of integration, one can find the probability for any intervals of interest. That is why function definition f(x) completely determines the distribution law for continuous random variables X.

For the probability distribution density f(x) must be satisfied normalization condition as:

f(x)dx = 1, (3.4)

if it is known that all values ​​of X lie in the interval (a, b), or in the form:

f(х) dх = 1, (3.5)

if the boundaries of the interval for the values ​​of X are not exactly known. Conditions for normalizing the probability density (3.4) or (3.5) are a consequence of the fact that the values ​​of the random variable X reliably lie within (a, b) or (-¥, +¥). From (3.4) and (3.5) it follows that the area of ​​the figure bounded by the distribution curve and the x-axis is always equal to 1.

The results presented in paragraphs 3.1 and 3.2 show that the laws of their distribution give a complete description of discrete or continuous random variables.

However, in many practically significant situations, the so-called numerical characteristics random variables, the main purpose of which is to express in a compressed form the most significant features of their distribution. It is important that these parameters are specific (constant) values, which can be estimated using the data obtained in the experiments. The so-called "Descriptive Statistics" deals with these estimates.

In probability theory and mathematical statistics, quite a lot of different characteristics are used, here we consider the most commonly used ones. Only for some of them are formulas used to calculate their values, in other cases we will leave the calculations to the computer.

3.3.1 Position characteristics: mathematical expectation, mode, median.

It is they that characterize the position of a random variable on the number axis, i.e., indicate some of its important values ​​that characterize the distribution of other values. Among them, the most important role is played by the mathematical expectation M(X).

A). Mathematical expectation M(X) random variable X is a probabilistic analogue of its arithmetic mean.

For a discrete random variable, it is calculated by the formula:

М(Х) = х1р1 + х2р2 + … + хnрn = = , (3.6)

and in the case of a continuous random variable M(X) are determined by the formulas:

M(X) = or M(X) = (3.7)

where f(x) is the probability density, dP= f(x)dx is the probability element (analogous to pi) for a small interval Dx (dx).

Example. Calculate the average value of a continuous random variable that has a uniform distribution on the interval (a, b).

Solution: With a uniform distribution, the probability density on the interval (a, b) is constant, i.e., f(x) = fo = const, and outside (a, b) it is equal to zero, and from the normalization condition (4.3) we find the value f0:

F0= f0 × x | = (b-a)f0 , whence

M(X) = | = = (a + b).

Thus, the mathematical expectation M(X) coincides with the middle of the interval (a, b), which determines , i.e. = M(X) = .

B). Mode Mo(X) of a discrete random variable call her most likely value(Fig. 3.4, a), and continuous- meaning X, at which density probabilities maximum(Fig. 3.4, b).

V). Another feature of the position is median (Me) distribution of a random variable.

median Fur) random variable is called its value X, which divides the entire distribution into two equiprobable parts. In other words, for a random variable equally probably accept values less Me (X) or more Me(X): P(X< Ме) = Р(Х >Me) = .

Therefore, the median can be calculated from the equation:

(3.8)

Graphically, the median is the value of a random variable whose ordinate divides square, limited by the distribution curve, in half (S1 \u003d S2) (Fig. 3.4, c). This feature is usually used only for continuous random variables, although it can be formally defined for discrete X.

If M(X), Mo(X) and Me(X) coincide, then the distribution of the random variable is called symmetrical, otherwise - asymmetrical.

Scattering characteristics– variance and standard deviation (root mean square deviation).

DispersionD (X) random variable X is defined as the mathematical expectation of the squared deviation of random X from its mathematical expectation M(X):

D (X) = M 2 , (3.9)

or D (X) = M (X2) - a)

Therefore, for discrete random variable, the dispersion is calculated by the formulas:

D(X) = [хi – М(Х)]2 pi, or D(X) = хi2 pi –

and for a continuous value distributed in the interval (a, b):

a for the interval (-∞,∞):

D (X) \u003d 2 f (x) dx, or D (X) \u003d x2 f (x) dx -

Dispersion characterizes the average dispersion, the dispersion of the values ​​of the random variable X relative to its mathematical expectation. The word "dispersion" itself means "scattering".

But the variance D(X) has the dimension of the square of a random variable, which is very inconvenient when estimating the spread in physical, biological, medical, and other applications. Therefore, another parameter is usually used, the dimension of which coincides with the dimension of X. This root mean square deviation random variable X, which is denoted s(X) :

s(X) = (3.13)

So, mean, mode, median, variance and standard deviation are the most used numerical characteristics of distributions of random variables, each of which, as has been shown, expresses some characteristic property of this distribution.

3.4. Normal law of distribution of random variables

Normal distribution law(Gauss's law) plays an extremely important role in probability theory. Firstly, this is the most common law of distribution of continuous random variables in practice. Secondly, he is limiting law, in the sense that other laws of distribution approach it under certain conditions.

normal law distribution is characterized by the following formula for the probability density:

, (3.13)

Here x are the current values ​​of the random variable X, and M(X) and s- its mathematical expectation and standard deviation, which completely determine the function f(x). Thus, if a random variable is distributed according to the normal law, then it is enough to know only two numerical parameters: M(X) and sto fully know the law of its distribution (3.13). The graph of the function (3.13) is called normal curve distribution(Gaussian curve). It has a symmetrical form with respect to the ordinate x = M(X). The maximum probability density, equal to ", corresponds to the mathematical expectation `X=M(X), and as you move away from it, the probability density f(x) decreases symmetrically, gradually approaching zero (Fig. Change in the value of M(X) in (3.13) does not change the shape of the normal curve, but only leads to its shift along the abscissa axis.The quantity M(X) is also called the scattering center, and the standard deviation s characterizes the width of the distribution curve (see Fig.3.6) .

With increasing s the maximum ordinate of the curve decreases, and the curve itself becomes flatter, stretching along the abscissa axis, whereas with a decrease s the curve stretches upward, simultaneously shrinking from the sides (Fig. 6).

Naturally, for any values ​​of M(X) and s, the area bounded by the normal curve and the X axis remains equal to 1 (normalization condition):

f(x) dx = 1, or f(x) dx =

The normal distribution is symmetrical, so M(X) = Mo(X) = Me(X).

The probability that the values ​​of the random variable X fall into the interval (x1,x2), i.e. P (x1< Х< x2) равна

R(x1< Х < x2) = . (3.15)

In practice, one often encounters the problem of finding the probability that the values ​​of a normally distributed random variable will fall into an interval that is symmetric with respect to M(X). In particular, let us consider the following problem, which is important from an applied point of view. Let us set aside segments equal to s, 2s and 3s from M(X) to the right and left (Fig. 7) and consider the result of calculating the probability of X falling into the corresponding intervals:

P (M(X) - s < Х < М(Х) + s) = 0,6827 = 68,27%. (3.16)

P (M(X) - 2s< Х < М(Х) + 2s) = 0,9545 = 95,45 %. (3.17)

P (M(X) - 3s< Х < М(Х) + 3s) = 0,9973 = 99,73 %. (3.18)

It follows from (3.18) that the values ​​of a normally distributed random variable X with parameters M(X) and s lie in the interval M(X) ± 3s with a probability Р = 99.73%, otherwise almost all possible values ​​of this random variable fall into this interval. quantities. This way of estimating the range of possible values ​​of a random variable is known as the "rule of three sigma".

Example. It is known that human blood pH is a normally distributed value with an average value (expectation) of 7.4 and a standard deviation of 0.2. Define the range of possible values ​​for this parameter.

Solution: To answer this question, we will use the “rule of three sigma”. With a probability equal to 99.73%, it can be argued that the range of pH values ​​for a person is 7.4 ± 3 0.2, i.e. 6.8 ÷ 8.

* If the exact values ​​of the boundaries of the interval are unknown, then consider the interval (-¥, + ¥).

ONE-DIMENSIONAL RANDOM VARIABLES

The concept of a random variable. Discrete and continuous random variables. Probability distribution function and its properties. Probability distribution density and its properties. Numerical characteristics of random variables: mathematical expectation, dispersion and their properties, standard deviation, mode and median; initial and central moments, asymmetry and kurtosis.

1. The concept of a random variable.

Random is called a quantity that, as a result of tests, takes one or another (but only one) possible value, known in advance, changing 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 are the size of a workpiece, the error in the result of measuring any parameter of a product or environment. Among the random variables encountered in practice, two main types can be distinguished: discrete variables and continuous ones.

Discrete is a random variable that takes on a finite or infinite countable set of values. For example, the frequency of hits with three shots; the number of defective products in a batch of pieces; the number of calls arriving at the telephone exchange during the day; the number of failures of the device elements for a certain period of time when testing it for reliability; the number of shots before 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, an error in measuring the range of a radar; chip uptime; manufacturing error of parts; salt concentration in sea water, etc.

Random variables are usually denoted by letters, etc., and their possible values ​​-, etc. To specify a random variable, it is not enough to list all its possible values. It is also necessary to know how often one or another of its values ​​​​may appear as a result of tests under the same conditions, i.e., it is necessary 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 a random variable.

2. Laws of distribution of a random variable.

distribution law A random variable is any correspondence between the 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 value has taken. Otherwise, random variables are called dependent. Several random variables are called mutually independent, if the distribution laws of any number of them do not depend on what possible values ​​the other quantities have taken.

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

The tabular assignment 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. In a graphical representation 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. Then build points and connect them with straight line segments. The resulting figure is called distribution polygon(Fig. 5). It should be remembered that the connection of the vertices of the ordinates is done only for the purposes of clarity, since in the intervals between and, and, etc., a random variable cannot take values, therefore the probabilities of its occurrence in these intervals are equal to zero.

The distribution polygon, like the distribution series, is one of the forms of specifying the distribution law of a discrete random variable. They can have very 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 a random variable, is always equal to one. This property follows from the fact that all possible values ​​of a random variable form a complete group of incompatible events, the sum of the probabilities of which is equal to one.

Definition. A random variable is a numerical value, the value of which depends on which elementary outcome occurred as a result of an experiment with a random outcome. The set of all values ​​that a random variable can take is called the set of possible values ​​for this random variable.

Random variables denote: X, Y 1, Z i; ξ , η 1, μ i, and their possible values ​​are x 3, y 1k, zij.

Example. In the experiment with a single throw of a dice, the random variable is the number X dropped points. Set of possible values ​​of a random variable X has the form

{x 1 \u003d 1, x 2 \u003d 2, ..., x 6 \u003d 6}.

We have the following correspondence between elementary outcomes ω and values ​​of the random variable X:

That is, for each elementary outcome ω i, i=1, …, 6, is assigned a number i.

Example. The coin is tossed until the first appearance of the "coat of arms". In this experiment, you can enter, for example, the following random variables: X- the number of throws before the first appearance of the "coat of arms" with many possible values ​​( 1, 2, 3, … ) And Y- the number of "digits" that fell out before the first appearance of the "coat of arms", with many possible values {0, 1, 2, …} (it's clear that X=Y+1). In this experiment, the space of elementary outcomes Ω can be identified with many

{G, CG, CG, …, C…CG, …},

and the elementary outcome ( Ts … TsG) is assigned to the number m+1 or m, Where m- the number of repetitions of the letter "C".

Definition. scalar function X(ω), given on the space of elementary outcomes, is called a random variable if for any x ∈ R (ω:X(ω)< x} is an event.

Distribution function of a random variable

To study the probabilistic properties of a random variable, it is necessary to know the rule that allows you to find the probability that a random variable will take a value from a subset of its values. Any such rule is called the law of probability distribution or the distribution of a random variable.

The general distribution law inherent in all random variables is the distribution function.

Definition. Distribution function (probabilities) of a random variable X call the function F(x), the value of which is at the point x equal to the probability of the event (X< x} , that is, an event consisting of those and only those elementary outcomes ω , for which X(ω)< x :

F(x) = P(X< x} .

It is usually said that the value of the distribution function at a point x is equal to the probability that the random variable X takes on a value less than x.

Theorem. The distribution function satisfies the following properties:

A typical form of the distribution function.

Discrete random variables

Definition. Random value X is called discrete if the set of its possible values ​​is finite or countable.

Definition. Near distribution (probabilities) of a discrete random variable X call a table consisting of two lines: the top line lists all possible values ​​​​of a random variable, and the bottom line lists the probabilities p i =P\(X=x i \) that the random variable takes these values.

To check the correctness of the table, it is recommended to sum the probabilities pi. By virtue of the axiom of normalization:

Based on the distribution series of a discrete random variable, one can construct its distribution function F(x). Let X- , given by its distribution series, and x 1< x 2 < … < x n . Then for all x ≤ x 1 event (X< x} is impossible, therefore, by definition F(x)=0. If x 1< x≤ x 2 , then the event (X< x} consists of those and only those elementary outcomes for which X(ω)=x 1. Hence, F(x)=p 1. Similarly, when x2< x ≤ x 3 event (X< x} consists of elementary outcomes ω , for which either X(ω)=x 1, or X(ω)=x2, that is (X< x}={X=x 1 }+{X=x 2 } . Hence, F(x)=p1 +p2 etc. At x > xn event (X< x} sure, then F(x)=1.

The distribution law of a discrete random variable can also be specified analytically in the form of some formula or graphically. For example, the distribution of a dice is described by the formula

P(X=i) = 1/6, i=1, 2, …, 6.

Some Discrete Random Variables

Binomial distribution. Discrete random variable X distributed according to the binomial law if it takes the values ​​0, 1, 2, ..., n in accordance with the distribution given by the Bernoulli formula:

This distribution is nothing but the distribution of the number of successes X V n tests according to the Bernoulli scheme with a probability of success p and failure q=1-p.

Poisson distribution. Discrete random variable X distributed according to the Poisson law if it takes non-negative integer values ​​with probabilities

Where λ > 0 is the Poisson distribution parameter.

The Poisson distribution is also called the law of rare events, since it always appears where a large number of trials are performed, in each of which a "rare" event occurs with a small probability.

In accordance with Poisson's law, distributed, for example, the number of calls received during the day at the telephone exchange; the number of meteorites that fell in a certain area; the number of decayed particles in the radioactive decay of matter.

Geometric distribution. Consider the Bernoulli scheme again. Let X is the number of trials to be done before the first success occurs. Then X- discrete random variable taking values ​​0, 1, 2, …, n, … Determine the probability of an event (X=n).

• X=0, if the first trial succeeds, therefore, P(X=0)=p.
• X=1 If the first trial fails and the second succeeds, then P(X=1)=qp.
• X=2, if in the first two trials - failure, and in the third - success, then P(X=2)=q 2 p.
• Continuing the procedure, we get P(X=i)=q i p, i=0, 1, 2, …

  A random variable with such a distribution series is called distributed according to a geometric law.