The random variable is called discrete, if the set of all its possible values is a finite or infinite, but necessarily countable set of values, i.e. such a set, all elements of which can be (at least theoretically) numbered and written out in the appropriate sequence.
Such random variables listed above, such as the number of points that fall out when throwing a dice, the number of pharmacy visitors during the day, the number of apples on a tree, are discrete random variables.
The most complete information about a discrete random variable is given by distribution law this value - it is a correspondence between all possible values of this random variable and their corresponding probabilities.
The law of distribution of a discrete random variable is often set in the form of a two-line table, the first line of which lists all possible values of this variable (in ascending order), and the second line lists the probabilities corresponding to these values:
| X | x 1 | x 2 | … | x n |
| P | p 1 | p 2 | … | p n |
Since all possible values of a discrete random variable represent a complete system, the sum of probabilities is equal to one ( normalization condition):
Example 4 There are ten student groups with 12, 10, 8, 10, 9, 12, 8, 11,10 and 9 students respectively. Write a distribution law for a random variable X, defined as the number of students in a randomly selected group.
Solution. The possible values of the considered random variable X (in ascending order) are 8, 9, 10, 11, 12. The probability that there will be 8 students in a randomly selected group is equal to
Similarly, you can find the probabilities of the remaining values of the random variable X:
Thus, the desired distribution law:
| X | |||||
| P | 0,2 | 0,2 | 0,3 | 0,1 | 0,2 |
The distribution law of a discrete random variable can also be specified using a formula that allows for each possible value of this variable to determine the corresponding probability (for example, the Bernoulli distribution, the Poisson distribution). To describe certain features of a discrete random variable, use its basic numerical characteristics: mathematical expectation, variance and standard deviation (standard).
mathematical expectation M (X) (the notation "μ" is also used) of a discrete random variable x is the sum of the products of each of all its possible values by the corresponding probabilities:
The main meaning of the mathematical expectation of a discrete random variable is that it represents mean given value. In other words, if a certain number of tests were performed, the results of which found the arithmetic mean of all observed values of a discrete random variable X, then this arithmetic mean is approximately equal (the more accurate, the greater the number of tests) to the mathematical expectation of this random variable.
Let us present some properties of mathematical expectation.
1. The mathematical expectation of a constant value is equal to this constant value:
M(S)=S
2. The mathematical expectation of the product of a constant factor by a discrete random variable is equal to the product of this constant factor by the mathematical expectation of a given random variable:
М(kX)=kM(X)
3. The mathematical expectation of the sum of two random variables is equal to the sum of the mathematical expectations of these variables:
M(X+Y)=M(X)+M(Y)
4. The mathematical expectation of the product of independent random variables is equal to the product of their mathematical expectations:
M(XY)=M(X)M(Y)
Separate values of a discrete random variable are grouped around the mathematical expectation as the center. To characterize the degree of spread of possible values of a discrete random variable relative to its mathematical expectation, the concept is introduced dispersion of a discrete random variable.
dispersionD(X) (the notation "σ 2 " is also used) of a discrete random variable X is the mathematical expectation of the square of the deviation of this quantity from its mathematical expectation:
D(X)=σ 2 =M((X - μ) 2),(11)
In practice, it is more convenient to calculate the variance by the formula
D (X) \u003d σ 2 \u003d M (X 2) - μ 2, (12)
Let us list the main properties of the dispersion.
- The dispersion of a constant value is zero:
- The variance of any random variable is a non-negative number:
D(X)≥0
- The variance of the product of a constant factor k by a discrete random variable is equal to the product of the square of this constant factor and the variance of the given random variable:
D(kX)=k 2 D(X).
In computational terms, it is not the variance that is more convenient, but another measure of the dispersion of a random variable X, which is the most commonly used standard deviation(standard deviation or simply standard).
Standard deviation discrete random variable is called Square root from its variance:
The convenience of the standard deviation is that it has the dimension of the random variable itself X, while the variance has a dimension representing the square of the dimension x.
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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 random value can take is called the set of possible values of this random variable.
Random variables denote: X, Y 1, Zi; 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 variable 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 a x 1< x≤ x 2 , then the event (X< x} consists of those and only those elementary outcomes for which X(ω)=x 1. Consequently, 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 } . Consequently, 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 in n Bernoulli trials with 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 it is produced. big number trials, in each of which there is a small probability that a "rare" event occurs.
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.
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 -, and 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 setting 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 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.
The simplest form of setting this law is a table that lists the possible values of a random variable and their corresponding probabilities.
Such a table is called the distribution series of the random variable X.
0 x 1 x 2 x 3 x 4 x 5 x 6
distribution function
The distribution law is a complete and exhaustive characteristic of a discrete random variable. However, it is not universal, since it cannot be applied to continuous random variables. A continuous random variable takes on an infinite number of values that fill a certain gap. It is practically impossible to compile a table that includes all the values of a continuous random variable. Therefore, for a continuous random variable, there is no distribution law, in the sense that it exists for a discrete random variable.
How to describe a continuous random variable?
For this, not the probability of the event X = x is used, but the probability of the event X<х, где х - некоторая переменная. Вероятность этого события зависит от х и является функцией х.
This function is called distribution function random variable X and is denoted F(x):
F(x)=P(X
The distribution function is a universal characteristic of a random variable. It exists for any random variables: discrete and continuous.
Distribution function properties:
1. When x 1 > x 2 F(x 1)> F(x 2)
2. F(-∞)=0
3. F(+∞)=1
The distribution function of a discrete random variable is a discontinuous step function, jumps occur at points corresponding to the possible values of the random variable, and the probabilities of these values are equal. The sum of these jumps is equal to one.
1 F(x)
 |
Numerical characteristics of random variables.
The main characteristics of discrete random variables are:
distribution function;
distribution range;
for a continuous random variable:
distribution function;
distribution density.
Any law represents some function, and the specification of this function completely describes the random variable.
However, when solving a number of practical problems, it is not always necessary to characterize a random variable in full. It suffices to indicate only some numerical parameters characterizing the random variable.
Such characteristics, the purpose of which is to represent in a concentrated form the most significant features of the distribution, are called numerical characteristics of a random variable.
Position Characteristics
(MOV, mode, median)
Of all the numerical characteristics of random variables used, the characteristics describing the position of the random variable on the numerical axis are more often used, namely, they indicate some average value around which the possible values of the random variable are grouped.
For this, they are used following characteristics:
· expected value;
the median.
The mathematical expectation (average value) is calculated as follows:
X 1 R 1 +x 2 R 2 +….+x n R n ∑ х i р i
р 1 + р 2 + …..+р n n
Given that ∑ p i , CAN equals M[X] = ∑ x i p i
The mathematical expectation of a random variable is the sum of the products of all possible values of a random variable and the probabilities of these values.
The above formulation is valid only for discrete random variables.
For continuous quantities
M[X] = ∫ x f(x)dx, where f(x) - distribution density X.
There are various ways to calculate the average. The most common forms of representing averages are arithmetic mean, median and mode.
The arithmetic mean is obtained by dividing the total value of a given attribute for the entire homogeneous statistical population by the number of units of this population. To calculate the arithmetic mean, the formula is used:
Хср = (Х1+Х2+... +Хn):n,
where Xi is the value of the attribute of the i-th unit of the population, n is the number of units of the population.
Fashion random variable is called its most probable value.
 |
M
 |
Median the value located in the middle of the ordered row is called. For an odd number of units in the series, the median is unique and is located exactly in the middle of the series; for an even number, it is defined as the average of two adjacent units of the population occupying a middle position.
Statistics is a branch of science that studies the quantitative side of mass phenomena public life consisting of separate elements, units. The combination of elements constitutes a statistical population. The purpose of the study is to establish quantitative patterns of development of this phenomenon. It is based on the application of probability theory and the law big numbers. The essence of this law lies in the fact that despite the individual random fluctuations of individual elements of the population, a certain regularity is manifested in the total mass, which is characteristic of this population as a whole. The greater the number of single elements characterizing the phenomenon under study is considered, the more clearly the regularity inherent in this phenomenon is revealed.
Crime is a social, mass phenomenon, it is a statistical set of numerous facts of single criminal manifestations. This gives grounds to apply the methods of the theory of statistics for its study.
In statistical studies of social phenomena, three stages can be distinguished:
1) statistical observation, i.e. collection of primary statistical material;
2) summary processing of the collected data, during which the results are calculated, summary (summary) indicators are calculated and the results are presented in the form of tables and graphs;
3) analysis, during which the regularities of the studied statistical population, the relationship between its various components are revealed, a meaningful interpretation of the generalizing indicators is carried out.
First step statistical research is a statistical observation. It plays a special role, since the errors made in the process of data collection are almost impossible to correct at further stages of work, which ultimately leads to incorrect conclusions about the properties of the phenomenon under study, their incorrect interpretation.
According to the method of recording facts, statistical observation is divided into continuous and discontinuous. Under continuous, or current, is understood such observation, in which the establishment and identification of facts is carried out as they arise. In discontinuous observation, facts are recorded either regularly at certain intervals, or as needed.
According to the coverage of the units of the surveyed population, continuous and non-continuous observation are distinguished. A continuous observation is one in which all units of the studied population are subject to accounting. So, for example, the registration of crimes is theoretically a continuous observation. However, in practice, a certain part of the crimes, called latent ones, remains outside the statistical population under study, and therefore, in fact, such an observation is not continuous. A discontinuous observation is one in which not all units of the studied population are subject to registration. It is divided into several types: observation of the main array, selective observation, and some others.
Observation of the main array (it is sometimes called the imperfect continuous method) is a type of non-continuous observation in which out of the entire set of units of the object, such a part of them is observed that constitutes the overwhelming, predominant share of the entire set. Observation by this method is practiced in those cases where the continuous coverage of all units of the population is associated with particular difficulties and at the same time, the exclusion from the observation of a certain number of units does not significantly affect the conclusions about the properties of the entire population. Therefore, the registration of crimes can rather be attributed to this type of observation.
Most perfect view non-continuous observation is selective, in which, in order to characterize the entire population, only a certain part of it is subjected to examination, however, taken on a sample according to certain rules. The main condition for the correctness of the sampling observation is such a selection, as a result of which the selected part of the units, according to all the characteristics to be studied, would accurately characterize the entire population as a whole. Most often, selective observation is used in the course of sociological research. In the future, we will consider the rules and methods for selecting units during selective observation.
After the primary material is collected and verified, the second stage of the statistical study is carried out - a summary. Statistical observation provides material that characterizes individual units of the object of study. The task of the summary is to summarize, systematize and summarize the results of the observation so that it becomes possible to identify character traits and essential properties, to discover the regularities of the studied phenomena and processes.
The simplest example of a summary is the summation of all reported crimes. However, such a generalization does not give a complete picture of all the properties of the criminogenic situation. In order to characterize crime deeply and comprehensively, it is necessary to know how the total number of crimes is distributed by type, time, place and method of committing, etc.
The distribution of units of the object under study into homogeneous groups according to their essential features is called statistical grouping. Objects studied by statistics are usually characterized by many properties and relationships expressed by various features. Therefore, the grouping of the examined objects can be carried out depending on the objectives of the statistical study according to one or more of these features. Thus, the personnel of the body can be grouped by positions, special ranks, age, length of service, marital status, etc.
As a result of processing and systematization of primary statistical materials, series of digital indicators are obtained that characterize certain aspects of the studied phenomena or processes or their change. These rows are called statistical. According to their content, statistical series are divided into two types: distribution series and dynamics series. Distribution series are series that characterize the distribution of units of the initial population according to any one attribute, the varieties of which are arranged in a certain order. For example, distributions total crimes into separate types, the number of all personnel by positions represent the distribution series.
Dynamic series are series that characterize the change in the size of social phenomena over time. A detailed consideration of such series and their use in the analysis and forecast of the criminal situation is the subject of a separate lecture.
The results of statistical observation and summaries of its materials are expressed primarily in absolute values (indicators). Absolute values show the dimensions of a social phenomenon under given conditions of place and time, for example, the number of crimes committed or the number of persons who committed them, the actual number of personnel or the number of vehicles. Absolute values are divided into individual and total (i.e. total). Absolute values are called individual, expressing the size of quantitative characteristics of individual units of a particular set of objects (for example, the number of victims or material damage in a particular criminal case, the age or length of service of a given employee, his salary, etc.). They are obtained directly in the process of statistical observation and are recorded in primary accounting documents. Individual absolute values are the basis of any statistical study.
In contrast to individual total absolute values, they characterize the final value of a feature for a certain set of objects covered by statistical observation. They are obtained either by directly counting the number of units of observation (for example, the number of crimes of a certain type), or as a result of summing the values of the attribute for individual units of the population (for example, the damage caused by all crimes).
However, absolute values, taken by themselves, do not always give a proper idea of the phenomena and processes under study. Therefore, along with the absolute values great importance in statistics have relative values.
Comparison is the main technique for evaluating statistical data and an integral part of all methods of their analysis. However, a simple comparison of two quantities is not enough to accurately assess their relationship. This ratio must also be measured. The role of the measure of such a ratio is performed by relative values.
Unlike absolute, relative values are derived indicators. They are obtained not as a result of simple summation, but by relative (multiple) comparison of absolute values.
Depending on the nature of the phenomenon under study and the specific objectives of the study, relative quantities can have a different form (appearance) of expression. The simplest form of expressing a relative value is a number (integer or fractional), showing how many times one value is greater than the other, taken as the basis of comparison, or what part of it is.
Most often, in the analytical activities of the internal affairs bodies, a different form of representation of relative numbers is used, a percentage, in which the main value is taken as 100. To determine the percentage, it is necessary to multiply the result of dividing one absolute value by another (base) by 100.
An important role in the summary processing of statistical data belongs to the average value. Since each individual unit of the statistical population has individual features, differing from any other quantitative value, to characterize the properties of the entire statistical population as a whole, average value . In statistics, the average value is understood as an indicator that reflects the level of a sign that changes in size per unit of a homogeneous population.
To characterize the homogeneity of the statistical population
various indicators are used according to the relevant attribute: variation, variance, standard deviation. These indicators make it possible to assess to what extent the corresponding average value reflects the properties of the entire population as a whole, whether it can generally be used as a generalizing characteristic of this statistical population. Detailed consideration of these indicators is an independent issue.
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Discrete random variables
Let some test be performed, the result of which is one of the incompatible random events (the number of events is either finite or countable, that is, the events can be numbered). Each outcome is assigned a certain real number, that is, a real function X with values is given on the set of random events. This X function is called discrete random magnitude(The term "discrete" is used because the values of a random variable are single numbers, as opposed to continuous functions). Since the values of a random variable change depending on random events, the main interest is the probabilities with which a random variable takes on different numerical values. The distribution law of a random variable is any relation that establishes a connection between the possible values of a random variable and their corresponding probabilities. The law of distribution can take various forms. For a discrete random variable, the distribution law is a set of pairs of numbers (), where are the possible values of the random variable, and are the probabilities with which it takes these values: . Wherein.
Pairs can be viewed as points in some coordinate system. By connecting these points with line segments, we get a graphic representation of the distribution law - the distribution polygon. Most often, the distribution law of a discrete random variable is written in the form of a table in which pairs are entered.
Example. The coin is flipped twice. Draw up the law of distribution of the number of "coats of arms" falling out in this test.
Solution. Random variable X - the number of "coat of arms" in this test. Obviously, X can take one of three values: 0, 1, 2. The probability of a “coat of arms” appearing in one toss of a coin is equal to p = 0.5, and a “tails” is q = 1 - p = 0.5. The probabilities with which the random variable takes on the listed values can be found using the Bernoulli formula:
We write the distribution law of a random variable X in the form of a distribution table
Control:
Some laws of distribution of discrete random variables, often encountered in solving various problems, have received special names: geometric distribution, hypergeometric distribution, binomial distribution, Poisson distribution, and others.
The distribution law of a discrete random variable can be specified using the distribution function F(x), which is equal to the probability that the random variable X will take values on the interval ????x?: F(x) = P(X The function F(x) is defined on the entire real axis and has the following properties: one) ? ? F(x)? one; 2) F(x) - non-decreasing function; 3) F(??) = 0, F(+?) = 1; 4) F(b) - F(a) = P(a ? X< b) - вероятность того, что случайная величина Х примет значения на промежутке 2 =(1-2.3) 2 =1.69 2 =(2-2.3) 2 =0.09 2 =(5-2.3) 2 =7.29 Let's write the distribution law of the squared deviation: Solution: Find the mathematical expectation M(x): M(x)=2*0.1+3*0.6+5*0.3=3.5 Let's write the distribution law of the random variable X 2 Let's find the mathematical expectation M(x 2): M(x 2)=4*0.1+9*0.6+25*0.3=13.5 The desired dispersion D (x) \u003d M (x 2) - 2 \u003d 13.3- (3.5) 2 \u003d 1.05 Dispersion Properties 1. The dispersion of the constant C is zero: D(C)=0 2. A constant factor can be taken out of the dispersion sign by squaring it. D(Cx)=C 2 D(x) 3. The variance of the sum of independent random variables is equal to the sum of the variances of these variables. D(X 1 +X 2 +...+X n)=D(X 1)+D(X 2)+...+D(X n) 4. The variance of the binomial distribution is equal to the product of the number of trials and the probability of occurrence and non-occurrence of an event in one trial D(X)=npq. To estimate the dispersion of possible values of a random variable around its mean value, in addition to the variance, some other characteristics also serve. Among them is the standard deviation. DEFINITION. The standard deviation of a random variable X is the square root of the variance: Example 8. Random variable X is given by the distribution law Find the standard deviation y(x) Solution: Find the mathematical expectation X: M(x)=2*0.1+3*0.4+10*0.5=6.4 Find the mathematical expectation X 2: M(x 2)=2 2 *0.1+3 2 *0.4+10 2 *0.5=54 Let's find the variance: D (x) \u003d M (x 2) \u003d M (x 2) - 2 \u003d 54-6.4 2 \u003d 13.04 Required standard deviation y(X)=vD(X)=v13.04?3.61 Theorem. The standard deviation of the sum of a finite number of mutually independent random variables is equal to the square root of the sum of the squared standard deviations of these variables: random variables The concept of a random variable is fundamental in probability theory and its applications. Random variables, for example, are the number of points dropped in a single throw of a dice, the number of decayed radium atoms in a given period of time, the number of calls at a telephone exchange in a certain period of time, the deviation from the face value of a certain size of a part with a properly established technological process, etc. In this way, random
magnitude A variable is called a variable that, as a result of experience, can take on one or another numerical value. In what follows, we will consider two types of random variables -- discrete and continuous. 1. Discrete random variables Consider a random variable * whose possible values form a finite or infinite sequence of numbers x1
,
x2
,
.
..,
xn,
.
..
.
Let the function p(x), whose value at each point x=xi(i=1,2,.
..)
is equal to the probability that the value will take on the value xi.
This random variable is called discrete
(intermittent). Function p(x) called law
distribution
probabilities
random
quantities, or briefly, law
distribution. This function is defined at the points of the sequence x1
,
x2
,
.
..,
xn,
.
..
.
Since in each of the tests the random variable always takes some value from the area of its change, then Example1.
Random variable - the number of points that drop out when a single throw of a dice. Possible values are numbers 1, 2, 3, 4, 5 and 6. Moreover, the probability that any of these values will take is the same and equal to 1/6. What will be the distribution law? ( Solution) Example2.
Let the random variable be the number of occurrence of the event A in one test, and P(A)=p. The set of possible values consists of 2 numbers 0 and 1: =0
if the event A did not happen and =1
if the event A happened. In this way, Let's assume that it is produced n independent tests, each of which may or may not result in an event A. Let the probability of an event A for each test is p A at n independent tests. The range consists of all integers from 0
before n inclusive. Probability distribution law p(m) is determined by the Bernoulli formula (13"): The law of probability distribution according to the Bernoulli formula is often called binomial, because Pn(m) represents m th term of the binomial expansion. Let the random variable take on any non-negative integer value, and where is some positive constant. In this case, the random variable is said to be distributed over law
Poisson, Note that when k=0 should be put 0!=1
. As we know, for large numbers n independent test probability Pn(m) offensive m event times A it is more convenient to find not by the Bernoulli formula, but by the Laplace formula [see. formula (15)]. However, the latter gives large errors with a low probability R occurrence of an event BUT in one test. In this case, to calculate the probability Pn(m) it is convenient to use the Poisson formula, in which The Poisson formula can be obtained as a limiting case of the Bernoulli formula with an unlimited increase in the number of trials n and as the probability tends to zero. Example3.
A batch of parts in the amount of 1000 pieces arrived at the plant. The probability that a part will be defective is 0.001. What is the probability that there will be 5 defective parts among the arrived parts? ( Solution) The Poisson distribution is often encountered in other problems as well. So, for example, if a telephone operator, on average, receives N calls, then, as can be shown, the probability P(k) that within one minute she will receive k calls, is expressed by the Poisson formula, if we put. If the possible values of a random variable form a finite sequence x1
,
x2
,
.
..,
xn, then the probability distribution law of the random variable is given in the form of the following table, in which Values Probabilities p(xi) This table is called beside
distribution random variable. Visually function p(x) can be shown as a graph. To do this, we take a rectangular coordinate system on the plane. We will plot the possible values of the random variable along the horizontal axis, and the values of the function along the vertical axis. Function Graph p(x) shown in fig. 2. If you connect the points of this graph with straight line segments, you get a figure called polygon
distribution. Example4.
Let the event BUT- the appearance of one point when throwing a dice; P(A)=1/6. Consider a random variable - the number of occurrences of an event BUT with ten throws of a dice. Function values p(x)(distribution law) are given in the following table: Values Probabilities p(xi) Probabilities p(xi)
calculated by the Bernoulli formula for n=10. For x>6 they are almost zero. The graph of the function p(x) is shown in fig. 3. The probability distribution function of a random variable and its properties Consider the function F(x), defined on the entire numerical axis as follows: for each X meaning F(x) is equal to the probability that a discrete random variable will take a value less than X, i.e. This function is called function
distribution
probabilities, or briefly, function
distribution. Example1.
Find the distribution function of the random variable given in Example 1, item 1. ( Solution) Example2.
Find the distribution function of the random variable given in Example 2, item 1. ( Solution) Knowing the distribution function F(x), it is easy to find the probability that a random variable satisfies the inequalities. Consider the event that a random variable takes on a value less than. This event breaks down into the sum of two incompatible events: 1) the random variable takes on values that are smaller, i.e. ; 2) the random variable takes values that satisfy the inequalities. Using the addition axiom, we get But by definition of the distribution function F(x)[cm. formula (18)], we have therefore, In this way, probability
hits
discrete
random
quantities
in
interval
is equal to
increment
functions
distribution
on the
this
interval. Considermainpropertiesfunctionsdistribution. 1°. Function
distribution
is
non-decreasing. Indeed, let< . Так как вероятность любого события неотрицательна, то. Поэтому из формулы (19) следует, что 2°. Values
functions
distribution
satisfy
inequalities . This property stems from the fact that F(x) defined as a probability [cf. formula (18)]. It is clear that * and. 3°. Probability
Togo,
what
discrete
random
magnitude
will accept
one
from
possible
values
xi,
is equal to
gallop
functions
distribution
in
point
xi. Indeed, let xi- the value taken by a discrete random variable, and. Assuming in formula (19) , we obtain In the limit at, instead of the probability that a random variable falls into the interval, we obtain the probability that the value will take on a given value xi: On the other hand, we get, i.e. function limit F(x) right, because. Therefore, in the limit formula (20) takes the form those. meaning p(xi)
equals function jump ** xi. This property is clearly illustrated in Fig. 4 and fig. 5. Continuous random variables In addition to discrete random variables, the possible values of which form a finite or infinite sequence of numbers that do not completely fill any interval, there are often random variables whose possible values form a certain interval. An example of such a random variable is the deviation from the nominal value of a certain size of a part with a properly established technological process. This kind of random variables cannot be specified using the probability distribution law p(x). However, they can be specified using the probability distribution function F(x). This function is defined in exactly the same way as in the case of a discrete random variable: Thus, here too the function F(x) defined on the whole number axis, and its value at the point X is equal to the probability that the random variable will take on a value less than X. Formula (19) and properties 1° and 2° are valid for the distribution function of any random variable. The proof is carried out similarly to the case of a discrete quantity. The random variable is called continuous, if for it there exists a non-negative piecewise-continuous function* that satisfies for any values x equality The function is called density
distribution
probabilities, or briefly, density
distribution. If a x 1
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 a 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 Consequently In this way, probability
Togo,
what
continuous
random
magnitude
maybe
to 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 sections are devoted to the distributions of continuous random variables that are frequently encountered in practice - the 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 (that is, it 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 assume 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 graph of the distribution function 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 function of the joint distribution 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 moments of the second order. 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 largely reduced 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. one). 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 with the help of 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. four). 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 considered examples, 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. Let us 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. four). 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. In this way. 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. random variables. Function and probability distribution density of a discrete random variable. Singular random variables. Mathematical expectation of a random variable. Chebyshev's inequality. Moments, cumulants and characteristic function. abstract, added 03.12.2007 Concepts of probability theory and mathematical statistics, their application in practice. Definition of a random variable. Types and examples of random variables. The law of distribution of a discrete random variable. Laws of distribution of a continuous random variable. abstract, added 10/25/2015 The probability of hitting a random variable X in a given interval. Plotting the distribution function of a random variable. Determination of the probability that a product chosen at random meets the standard. The law of distribution of a discrete random variable. test, added 01/24/2013 Discrete random variables and their distributions. Total probability formula and Bayes formula. General properties of mathematical expectation. Dispersion of a random variable. Distribution function of a random variable. The classical definition of probabilities. control work, added 12/13/2010 Distribution function of a continuous random variable. The mathematical expectation of a continuous random variable, the probability distribution density of the system. covariance. Correlation coefficient. laboratory work, added 08/19/2002 Features of the distribution function as the most universal characteristic of a random variable. Description of its properties, their representation with the help of geometric interpretation. Patterns of calculating the probability of distribution of a discrete random variable. presentation, added 11/01/2013 Determining the probabilities of various events using the Bernoulli formula. Compilation of the law of distribution of a discrete random variable, calculation of mathematical expectation, variance and standard deviation of a random variable, probability densities. control work, added 10/31/2013 Using the Bernoulli formula to find the probability of an event occurring. Plotting a discrete random variable. Mathematical expectation and properties of the integral distribution function. Distribution function of a continuous random variable. test, added 01/29/2014 Theory of Probability and Regularities of Mass Random Phenomena. Inequality and Chebyshev's theorem. Numerical characteristics of a random variable. Distribution density and Fourier transform. Characteristic function of a Gaussian random variable. abstract, added 01/24/2011 Calculation of mathematical expectation, variance, distribution function and standard deviation of a random variable. The law of distribution of a random variable. The classic definition of the probability of an event. Finding the distribution density.
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