Using the considered discrete random variables, it is impossible to describe real random experiments. Indeed, such quantities as the size of any physical objects, temperature, pressure, the duration of certain physical processes, cannot be assigned a discrete set of possible values. It is natural to assume that this set fills some numerical interval. Therefore, the concept of a continuous random variable.
A continuous random variable is a random variable X, the set of values of which is a certain numerical interval.
Consider examples of continuous random variables.
1. X - the time interval between two failures (failures) of a computer. Then 
.
2. X - the height of the rise of water in the flood. In this case 
.
It is clear that for a continuous random variable, the values of which completely fill a certain interval of the abscissa axis, it is impossible to construct a distribution series. First, you can't list one after the other. possible values and secondly, as we will show below, the probability of a single value of a continuous random variable is equal to zero.
Otherwise, i.e. if each individual value of a continuous random variable were assigned a non-zero probability, then when summing up all the probabilities, one could get a number other than one, since the set of values of a continuous random variable is uncountable (the values fill entirely a certain interval).
Let the set contain an uncountable set of values of a continuous random variable X. The system of subsets is formed by any subsets that can be obtained from the set , ,
by applying a countable number of times the operations of union, intersection, addition. System ,
therefore, will contain sets of the form ( x 1<Х<х 2
} ,
, 
, 
, , , .
To define a probability measure on these sets, we introduce the concept of a probability distribution density.
Definition 2.5. The probability distribution density p(x) of a continuous random variable X is the limit, if it exists, of the ratio of the probability of the random variable X falling into the interval adjacent to the point x to the length of this interval when the latter tends to zero i.e.
(2.4)
The curve depicting the probability distribution density (probability density) of a continuous random variable is called the distribution curve. For example, the distribution curve may look like in Fig. 2.4.
It should be noted that if p(x) multiply by , then the value p(x) called element of probability characterizes the probability that X takes values from an interval of length adjacent to the point X. Geometrically, this is the area of a rectangle with sides and p(x)(see fig. 2.4 ).
Then the probability of hitting a continuous random variable X on the segment will be equal to the sum of the probability elements on this entire segment, i.e. area of a curvilinear trapezoid bounded by a curve y=p(x), axis Oh and direct X = a, x = β:
, (2.5)
since the area of the shaded figure will tend to the area of the curvilinear trapezoid at (Fig. 2.5).
The probability density has the following properties.
1
°. p(x) 0 
, since the limit of non-negative quantities is a non-negative quantity.
2
°. 
,
since the probability that a continuous random variable takes values from the interval , i.e. the probability of a certain event is equal to one.
3 °. p(x) is continuous or piecewise continuous.
Thus, formula (2.5) introduces a normalized probability measure on any subsets of the set .
Distribution function of a random variable X - this is a function F(x) real variable X, which determines the probability that a random variable takes values less than some fixed number X, those. 
: .
Then it follows from formula (2.5) that for any
. (2.6)
Geometrically, the distribution function is the area of the figure lying to the left of the point X, limited distribution curve at= p(x) and the abscissa axis. From formula (2.6) and Barrow's theorem for the case when p(x) continuous, it follows that
p(x) = (2.7)
Fig.2.6 Fig.2.7
This equality is violated at the discontinuity points of the probability density. Schedule F(x) continuous random variable X may look like the curve shown in Fig. 2.6.
Let us give a rigorous definition of a continuous random variable.
Definition 2.6.A random variable X is called continuous if there exists a non-negative function p(x) such that equality (2.6) holds for any.
distribution function F(x), satisfying equality (2.6) is called absolutely continuous.
So, the distribution function of a continuous random variable defines an absolutely continuous distribution of a random variable.
For a continuous random variable X the following theorem holds.
Theorem 2.4. The probability of a single value of a continuous random variable X is zero:
Proof. By Theorem 2.3, the probability of a single value is:
Since for a continuous random variable , then .
From the proved theorem follows the validity of the equalities:
Indeed, since 
etc.
Thus, to calculate the probabilities of arbitrary events , where it is necessary to set on the set of values of a continuous random variable or the distribution function F(x), or the probability distribution density p(x).
Example 2.4. Random value X has a probability distribution density
Find parameter With and distribution function F(x). Plot function graphs p(x) And F(x).
Solution. To find the parameter With, use the property 2
○ probability distribution density: . Substituting the density value, we get 
. Having calculated the integral 
, find the value of c from the equality: , .
The probability distribution density will take the form
Since the density is given using three formulas, the calculation of the distribution function depends on the location on the real axis. If:
1) , then using formula (2.6), we obtain
Above, a continuous random variable was specified using a distribution function. This way of setting is not the only one. A continuous random variable can also be specified using a function called distribution density or probability density (often called differential function ).
The probability distribution density of a continuous random variable X call the function f(x)- first derivative of the distribution function F(x):
f(x)=F"(x).
From this definition it follows that the distribution function is primitive for the distribution density. Knowing the distribution density, we can calculate the probability that a continuous random variable will take a value that belongs to a given interval.
Theorem. The probability that a continuous random variable X will take a value belonging to the interval ( a, b), is equal to a certain integral of the distribution density, taken in the range from A before b:
Knowing the distribution density f(x), we can find the distribution function F(x) according to the formula
.
Distribution density properties:
Property 1. The distribution density is a non-negative function: 
.
Geometrically, this property means that the points belonging to the distribution density plot are located either above the axis Oh, or on this axis. The plot of the distribution density is called distribution curve .
Property 2. Improper integral of the distribution density ranging from 
before 
is equal to one:
.
Geometrically, this means that the entire area of the curvilinear trapezoid bounded by the Ox axis and the distribution curve is equal to one.
In particular, if all values of the random variable belong to the interval ( a, b), That
.
Mathematical expectation of a discrete random variable
The distribution law fully characterizes the random variable. However, it is often unknown in advance and one has to use indirect information. In many cases, these indirect characteristics are quite sufficient for solving practical problems and it is not necessary to determine the distribution law. Such characteristics are called numerical characteristics ticks of a random value. And the first of them is the mathematical expectation.
Mathematical expectation of a discrete random variable X is the sum of the products of all its possible values ( x 1 , x 2 , …, x n) on their probabilities ( p 1 , p 2 , …, p n):
It should be noted that M(x) There is nonrandom (constant. It can be proved that M(x) is approximately equal (and the more accurate, the greater the number of trials n) to the arithmetic mean of the observed values of the random variable.
The mathematical expectation has the following properties:
· Expected value constant equal to the most constant:
.
· Constant multiplier can be taken out of the mathematical expectation sign:
.
· Expected value works two independent random variables X And Y(that is, the distribution law of one of them does not depend on the possible values of the other) is equal to the product of their mathematical expectations:
· Expected value amounts of two random variables is equal to the sum of the mathematical expectations of the terms:
Here under sum X+Y random variables is understood as a new random variable, the values of which are equal to the sums of each value X with every possible value Y; probabilities of possible values X+Y for independent random variables X And Y are equal to the products of the probabilities of the terms, and for dependent ones, to the products of the probabilities of one term and the conditional probability of the other. So if X And Y are independent and their distribution laws
If produced n independent trials,
each of which the probability of an event A constant and equal p, then the mathematical expectation number of appearances events A in series:
.
Note that the third and fourth properties are easily generalized for any number of random variables.
Dispersion of a discrete random variable
The mathematical expectation is a convenient characteristic, but often it is not enough for judging the possible values of a random variable or how they scattered around the mean. Therefore, other numerical characteristics are also introduced.
Let X is a random variable with mathematical expectation M(X). deviation X 0 is the difference between a random variable and its mathematical expectation:
.
Mathematical expectation of deviation M(X 0) = 0.
Example. Let the law of distribution of quantity X:
The deviation is an intermediate characteristic, on the basis of which we will introduce a more convenient characteristic. dispersion (dispersion ) Discrete random variable is called the mathematical expectation of the squared deviation of the random variable:
For example, let's find the variance of the quantity X with the following distribution law:
Here . Required variance:
The value of the dispersion is determined not only by the values of the random variable, but also by their probabilities. Therefore, if two random variables have the same or close mathematical expectations (this is quite common), then the variances are usually different. This allows us to additionally characterize the random variable under study.
We list the properties of the dispersion:
Dispersion constant value is zero:
.
· Constant multiplier can be taken out of the dispersion sign by squaring it:
.
Dispersion amounts And differences two independent random variables is equal to the sum of the variances of these variables:
Dispersion number of appearances events A V n independent tests, in each of which the probability P occurrence of an event constant , is determined by the formula:
,
Where 
is the probability that the event will not occur.
A convenient auxiliary characteristic used in calculations even more often than D(X), is standard deviation (or standard ) random variable:
.
The fact is that D(X) has the dimension of the square of the dimension of the random variable, and the dimension of the standard X) is the same as for the random variable X. This is very convenient for estimating the spread of a random variable.
Example. Let the random variable be given by the distribution:
| X | 2m | 3m | 10m |
| P | 0,1 | 0,4 | 0,5 |
Calculate: m
and standard: m.
Therefore, about a random variable X one can say either - its mathematical expectation is 6.4 m with a dispersion of 13.04 m 2, or - its mathematical expectation is 6.4 m with a dispersion 
m. The second formulation is obviously clearer.
Note that for the sum n independent random variables:
Initial and central theoretical moments
For most practical calculations of the numerical characteristics introduced above MX),DX)and X) enough. However, to study the behavior of random variables, you can also use some additional numerical characteristics that allow you to track the nuances of the behavior of a random variable and generalize the above theory.
The initial moment of the kth order of the random variable X is called the mathematical expectation of the quantity X k :
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The result of any random experiment can be characterized qualitatively and quantitatively. Qualitative the result of a random experiment - random event. Any quantitative characteristic, which as a result of a random experiment can take one of a certain set of values, - random value. Random value is one of the central concepts of probability theory.
Let be an arbitrary probability space. Random variable is a real numerical function x \u003d x (w), w W , such that for any real x 
.
Event usually written as x< x. In the following, random variables will be denoted by lowercase Greek letters x, h, z, …
A random variable is the number of points that fell when throwing a dice, or the height of a student randomly selected from the study group. In the first case, we are dealing with discrete random variable(it takes values from a discrete number set M=(1, 2, 3, 4, 5, 6) ; in the second case, with continuous random variable(it takes values from a continuous number set - from the interval of the number line I=).
Each random variable is completely determined by its distribution function.
If x . is a random variable, then the function F(x) = F x(x) = P(x< x) is called distribution function random variable x . Here P(x<x) - the probability that the random variable x takes a value less than x.
It is important to understand that the distribution function is a "passport" of a random variable: it contains all the information about the random variable and therefore the study of a random variable consists in the study of its distribution functions, often referred to simply distribution.
The distribution function of any random variable has the following properties:
If x is a discrete random variable taking the values x 1 <x 2 < … <x i < … с вероятностями p 1 <p 2 < … <pi < …, то таблица вида
| x 1 | x 2 | … | x i | … |
| p 1 | p 2 | … | pi | … |
called distribution of a discrete random variable.
The distribution function of a random variable with such a distribution has the form
A discrete random variable has a stepwise distribution function. For example, for a random number of points that fell out in one throw of a dice, the distribution, distribution function and distribution function graph look like:
| 1 | 2 | 3 | 4 | 5 | 6 |
| 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |
If the distribution function F x(x) is continuous, then the random variable x is called continuous random variable.
If the distribution function of a continuous random variable differentiable, then a more visual representation of the random variable gives probability density of random variable p x(x), which is related to the distribution function F x(x) formulas
And 
.
From this, in particular, it follows that for any random variable .
When solving practical problems, it is often necessary to find the value x, at which the distribution function F x(x) random variable x takes a given value p, i.e. you need to solve the equation F x(x) = p. Solutions to such an equation (the corresponding values x) in probability theory are called quantiles.
Quantile x p ( p-quantile, level quantile p) a random variable having a distribution function F x(x), is called the solution xp equations F x(x) = p, p(0, 1). For some p the equation F x(x) = p may have several solutions, for some - none. This means that for the corresponding random variable, some quantiles are not uniquely defined, and some quantiles do not exist.
Let $X$ be a continuous random variable with a probability distribution function $F(x)$. Recall the definition of the distribution function:
Definition 1
A distribution function is a function $F(x)$ satisfying the condition $F\left(x\right)=P(X
Since the random variable is continuous, then, as we already know, the probability distribution function $F(x)$ will be a continuous function. Let $F\left(x\right)$ be also differentiable on the entire domain of definition.
Consider the interval $(x,x+\triangle x)$ (where $\triangle x$ is the increment of $x$). On him
Now, letting the values of the increment $\triangle x$ tend to zero, we get:
Picture 1.
Thus, we get:
The distribution density, like the distribution function, is one of the forms of the distribution law of a random variable. However, the distribution law can be written in terms of the distribution density only for continuous random variables.
Definition 3
The distribution curve is a graph of the function $\varphi \left(x\right)$, the distribution density of a random variable (Fig. 1).
Figure 2. Distribution density plot.
Geometric sense 1: The probability that a continuous random variable falls into the interval $(\alpha ,\beta)$ is equal to the area of the curvilinear trapezoid bounded by the distribution function graph $\varphi \left(x\right)$ and the straight lines $x=\alpha ,$ $x=\beta $ and $y=0$ (Fig. 2).
Figure 3. Geometric representation of the probability of a continuous random variable falling into the interval $(\alpha ,\beta)$.
Geometric sense 2: The area of an infinite curvilinear trapezoid bounded by the graph of the distribution function $\varphi \left(x\right)$, the line $y=0$ and the variable line $x$ is nothing but the distribution function $F(x)$ (Fig. 3 ).
Figure 4. Geometric representation of the probability function $F(x)$ in terms of the distribution density $\varphi \left(x\right)$.
Example 1
Let the distribution function $F(x)$ of the random variable $X$ have the following form.
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