Even distribution. continuous value X is evenly distributed on the interval ( a, b) if all its possible values are in this interval and the probability distribution density is constant:
For a random variable X, uniformly distributed in the interval ( a, b) (Fig. 4), the probability of falling into any interval ( x 1 , x 2 ) lying inside the interval ( a, b), is equal to:
(30)
Rice. 4. Graph of the uniform distribution density
Rounding errors are examples of uniformly distributed quantities. So, if all tabular values of a certain function are rounded to the same digit, then choosing a tabular value at random, we consider that the rounding error of the selected number is a random variable uniformly distributed in the interval
exponential distribution. Continuous random variable X It has exponential distribution
(31)
The graph of the probability distribution density (31) is shown in fig. 5.
Rice. 5. Graph of the density of the exponential distribution
Time T failure-free operation of a computer system is a random variable that has an exponential distribution with the parameter λ
, the physical meaning of which is the average number of failures per unit time, not counting system downtime for repair.
Normal (Gaussian) distribution. Random value X It has normal (gaussian) distribution, if the density distribution of its probabilities is determined by the dependence:
(32)
Where m = M(X) , .
At the normal distribution is called standard.
The graph of the density of the normal distribution (32) is shown in fig. 6.
Rice. 6. Graph of the density of the normal distribution
The normal distribution is the most common distribution in various random phenomena of nature. Thus, errors in the execution of commands by an automated device, errors in the launch of a spacecraft to a given point in space, errors in the parameters of computer systems, etc. in most cases have a normal or close to normal distribution. Moreover, random variables formed by the summation of a large number of random terms are distributed almost according to the normal law.
Gamma distribution. Random value X It has gamma distribution, if the density distribution of its probabilities is expressed by the formula:
(33)
Where 
is the Euler gamma function.
Chapter 6. Continuous random variables.
§ 1. Density and distribution function of a continuous random variable.
The set of values of a continuous random variable is uncountable and usually represents some finite or infinite interval.
A random variable x(w) given in a probability space (W, S, P) is called continuous(absolutely continuous) W if there exists a non-negative function such that, for any x, the distribution function Fx(x) can be represented as an integral
The function is called a function probability distribution density.
The properties of the distribution density function follow from the definition:
1..gif" width="97" height="51">
3. At points of continuity, the distribution density is equal to the derivative of the distribution function: .
4. The distribution density determines the distribution law of a random variable, since it determines the probability of a random variable falling into the interval:
5. The probability that a continuous random variable will take a specific value is zero: . Therefore, the following equalities are true:
The plot of the distribution density function is called distribution curve, and the area bounded by the distribution curve and the x-axis is equal to one. Then, geometrically, the value of the distribution function Fx(x) at the point x0 is the area bounded by the distribution curve and the x-axis and lying to the left of the point x0.
Task 1. The density function of a continuous random variable has the form:
Determine the constant C, construct the distribution function Fx(x) and calculate the probability .
Solution. The constant C is found from the condition We have:
whence C=3/8.
To construct the distribution function Fx(x), note that the interval divides the range of the x argument (the number axis) into three parts: https://pandia.ru/text/78/107/images/image017_17.gif" width="264 "height="49">
since the density x on the semiaxis is zero. In the second case
Finally, in the last case, when x>2,
Since the density vanishes on the semiaxis . So, the distribution function is obtained
Probability calculate by the formula . Thus,
§ 2. Numerical characteristics of a continuous random variable
Expected value for continuously distributed random variables is determined by the formula https://pandia.ru/text/78/107/images/image028_11.gif" width="205" height="56 src=">,
if the integral on the right converges absolutely.
Dispersion x can be calculated using the formula 
, and also, as in the discrete case, according to the formula https://pandia.ru/text/78/107/images/image031_11.gif" width="123" height="49 src=">.
All the properties of expectation and variance given in Chapter 5 for discrete random variables are also valid for continuous random variables.
Task 2. For a random variable x from Problem 1, calculate the mathematical expectation and variance .
Solution.
And that means
https://pandia.ru/text/78/107/images/image035_9.gif" width="184" height="69 src=">
For a graph of the uniform distribution density, see fig. .
Fig.6.2. Distribution function and distribution density. uniform law
The distribution function Fx(x) of a uniformly distributed random variable is
Fx(x)= 
Mathematical expectation and dispersion; .
The exponential (exponential) distribution. A continuous random variable x that takes non-negative values has an exponential distribution with parameter l>0 if the probability distribution density of the random variable is equal to
px(x)= 
Rice. 6.3. Distribution function and distribution density of the exponential law.
The distribution function of the exponential distribution has the form
Fx(x)=https://pandia.ru/text/78/107/images/image041_8.gif" width="17" height="41">.gif" width="13" height="15"> and , if its distribution density is equal to
.
The set of all random variables distributed according to the normal law with parameters and parameters is denoted by .
The distribution function of a normally distributed random variable is
.
Rice. 6.4. Distribution function and distribution density of the normal law
Normal distribution parameters are the mathematical expectation https://pandia.ru/text/78/107/images/image048_6.gif" width="64 height=24" height="24">
In the particular case when https://pandia.ru/text/78/107/images/image050_6.gif" width="44" height="21 src="> normal distribution is called standard, and the class of such distributions is designated https://pandia.ru/text/78/107/images/image052_6.gif" width="119" height="49">,
while the distribution function
Such an integral cannot be calculated analytically (it is not taken in “quadratures”), and therefore tables are compiled for the function. The function is related to the Laplace function introduced in Chapter 4
,
the following relation 
. In the case of arbitrary values of the parameters https://pandia.ru/text/78/107/images/image043_5.gif" width="21" height="21 src="> the random variable distribution function is related to the Laplace function using the relation:
.
Therefore, the probability of a normally distributed random variable falling into an interval can be calculated by the formula
.
A non-negative random variable x is called log-normally distributed if its logarithm h=lnx obeys the normal law. The mathematical expectation and variance of a log-normally distributed random variable are Mx= and Dx=.
Task 3. Let a random value be given https://pandia.ru/text/78/107/images/image065_5.gif" width="81" height="23">.
Solution. Here and https://pandia.ru/text/78/107/images/image068_5.gif" width="573" height="45">
Laplace distribution is set by the function fx(x)=https://pandia.ru/text/78/107/images/image070_5.gif" width="23" height="41"> and the kurtosis is gx=3.
Fig.6.5. Laplace distribution density function.
The random variable x is distributed over Weibull law, if it has a distribution density function equal to https://pandia.ru/text/78/107/images/image072_5.gif" width="189" height="53">
The Weibull distribution obeys the times of failure-free operation of many technical devices. In tasks of this profile, an important characteristic is the failure rate (mortality rate) l(t) of the studied elements of age t, determined by the relation l(t)=. If a=1, then the Weibull distribution turns into an exponential distribution, and if a=2 - into the so-called distribution Rayleigh.
Mathematical expectation of the Weibull distribution: -https://pandia.ru/text/78/107/images/image075_4.gif" width="219" height="45 src=">, where Г(а) is the Euler function. .
In various problems of applied statistics, so-called "truncated" distributions are often encountered. For example, the tax authorities are interested in the distribution of income of those persons whose annual income exceeds a certain threshold c0 established by taxation laws. These distributions turn out to be approximately the same as the Pareto distribution. Pareto distribution given by functions
Fx(x)=P(x
Here https://pandia.ru/text/78/107/images/image081_4.gif" width="60" height="21 src=">.
Task 4. The random variable is uniformly distributed on the interval . Find the density of a random variable .
Solution. It follows from the condition of the problem that
Next, the function is a monotonic and differentiable function on the interval and has an inverse function 
, whose derivative is equal Therefore,
§ 5. A pair of continuous random variables
Let two continuous random variables x and h be given. Then the pair (x, h) determines a "random" point on the plane. A pair (x, h) is called random vector or two-dimensional random variable.
joint distribution function random variables x and h and the function is called F(x, y)=Phttps://pandia.ru/text/78/107/images/image093_3.gif" width="173" height="25">. joint density the probability distribution of random variables x and h is a function such that 
.
The meaning of this definition of the joint distribution density is as follows. The probability that a “random point” (x, h) will fall into an area on a plane is calculated as the volume of a three-dimensional figure - a “curved” cylinder bounded by the surface https://pandia.ru/text/78/107/images/image098_3. gif" width="211" height="39 src=">
The simplest example of a joint distribution of two random variables is the two-dimensional uniform distribution on the setA. Let a bounded set M with area be given. It is defined as the distribution of the pair (x, h) given by the following joint density:
Task 5. Let a two-dimensional random vector (x, h) be uniformly distributed inside the triangle . Calculate the probability of inequality x>h.
Solution. The area of the indicated triangle is equal to (see Fig. No.?). By virtue of the definition of a two-dimensional uniform distribution, the joint density of random variables x, h is equal to
The event matches the set 
on a plane, that is, a half-plane. Then the probability
On the half-plane B, the joint density is equal to zero outside the set https://pandia.ru/text/78/107/images/image102_2.gif" width="15" height="17">. Thus, the half-plane B is divided into two sets and https://pandia.ru/text/78/107/images/image110_1.gif" width="17" height="23"> and , and the second integral is zero, since the joint density is zero there. That's why
If the joint distribution density for the pair (x, h) is given, then the densities and components x and h are called private densities and are calculated by the formulas:
https://pandia.ru/text/78/107/images/image116_1.gif" width="224" height="23 src=">
For continuously distributed random variables with densities px(x), ph(y), independence means that
Task 6. Under the conditions of the previous problem, determine whether the components of the random vector x and h are independent?
Solution. Let us calculate the partial densities and . We have:
https://pandia.ru/text/78/107/images/image119_1.gif" width="283" height="61 src=">
Obviously, in our case https://pandia.ru/text/78/107/images/image121_1.gif" width="64" height="25"> is the joint density of x and h, and j(x, y) is a function of two arguments, then
https://pandia.ru/text/78/107/images/image123_1.gif" width="184" height="152 src=">
Task 7. In the conditions of the previous problem, calculate .
Solution. According to the above formula, we have:
.
Representing the triangle as
https://pandia.ru/text/78/107/images/image127_1.gif" width="479" height="59">
§ 5. Density of the sum of two continuous random variables
Let x and h be independent random variables with densities https://pandia.ru/text/78/107/images/image128_1.gif" width="43" height="25">. The density of the random variable x + h is calculated from formula convolutions
https://pandia.ru/text/78/107/images/image130_0.gif" width="39" height="19 src=">. Calculate sum density.
Solution. Since x and h are distributed according to the exponential law with the parameter , their densities are equal to
Hence,
https://pandia.ru/text/78/107/images/image134_0.gif" width="339 height=51" height="51">
If x<0, то в этой формуле аргумент https://pandia.ru/text/78/107/images/image136_0.gif" width="65" height="25">is negative, and therefore . Therefore, if https://pandia.ru/text/78/107/images/image140_0.gif" width="359 height=101" height="101">
Thus, we got the answer:
https://pandia.ru/text/78/107/images/image142_0.gif" width="40" height="41 "> is normally distributed with parameters 0 and 1. Random variables x1 and x2 are independent and have normal distributions with parameters a1 and a2 respectively Prove that x1 + x2 has a normal distribution Random variables x1, x2, ... xn are distributed and independent and have the same distribution density function
.
Find the distribution function and distribution density of quantities:
a) h1 = min (x1 , x2, ...xn) ; b) h(2) = max(x1,x2, ... xn )
Random variables x1, x2, ... xn are independent and uniformly distributed on the interval [а, b]. Find distribution functions and distribution density functions of quantities
x(1) = min(x1,x2, ... xn) and x(2)= max(x1, x2, ...xn).
Prove that M https://pandia.ru/text/78/107/images/image147_0.gif" width="176" height="47">.
The random variable is distributed according to the Cauchy law. Find: a) the coefficient a; b) distribution function; c) the probability of hitting the interval (-1, 1). Show that the expectation of x does not exist. The random variable obeys the Laplace law with the parameter l (l>0): Find the coefficient a; build graphs of distribution density and distribution function; find Mx and Dx; find the probabilities of events (|x|< и {çxç<}. Случайная величина x подчинена закону Симпсона на отрезке [-а, а], т. е. график её плотности распределения имеет вид:
Write a formula for the distribution density, find Mx and Dx.
Computing tasks.
A random point A has a uniform distribution in a circle of radius R. Find the mathematical expectation and variance of the distance r of a point to the center of the circle. Show that the quantity r2 is uniformly distributed on the segment . 
The distribution density of a random variable has the form: 
Calculate the constant C, the distribution function F(x), and the probability 
The distribution density of a random variable has the form: 
Calculate the constant C, the distribution function F(x), and the probability 
The distribution density of a random variable has the form: 
Calculate constant C, distribution function F(x), variance and probability Random variable has distribution function 
Calculate the density of a random variable, the mathematical expectation, variance and probability Check that the function = 
can be a distribution function of a random variable. Find the numerical characteristics of this quantity: Mx and Dx. The random variable is uniformly distributed on the segment . Write out the distribution density. Find the distribution function. Find the probability of hitting a random variable on the segment and on the segment . The distribution density x is
.
Find the constant c, the distribution density h = and the probability
P (0.25 The computer uptime is distributed according to an exponential law with the parameter l = 0.05 (failures per hour), i.e. it has a density function p(x) = The solution of a certain problem requires trouble-free operation of the machine for 15 minutes. If a failure occurs during the solution of the problem, then the error is detected only at the end of the solution, and the problem is solved again. Find: a) the probability that no failure will occur during the solution of the problem; b) the average time for which the problem will be solved. A rod of length 24 cm is broken into two parts; we will assume that the break point is distributed uniformly along the entire length of the rod. What is the average length of most of the rod? A piece of length 12 cm is randomly cut into two parts. The cut point is evenly distributed along the entire length of the segment. What is the average length of a small part of the segment? The random variable is uniformly distributed on the interval . Find the distribution density of a random variable a) h1 = 2x + 1; b) h2 = -ln(1-x); c) h3 = . Show that if x has a continuous distribution function F(x) = P(x Find the density function and the distribution function of the sum of two independent quantities x and h with uniform distribution laws on the intervals and, respectively. The random variables x and h are independent and uniformly distributed on the intervals and, respectively. Calculate the density of the sum x+h. The random variables x and h are independent and uniformly distributed on the intervals and, respectively. Calculate the density of the sum x+h. The random variables x and h are independent and uniformly distributed on the intervals and, respectively. Calculate the density of the sum x+h. Random variables are independent and have an exponential distribution with density Let a continuous random variable X be given by the distribution function f(x). Let us assume that all possible values of the random variable belong to the interval [ a,b]. Definition. mathematical expectation continuous random variable X, the possible values of which belong to the segment , is called a definite integral If the possible values of a random variable are considered on the entire number axis, then the mathematical expectation is found by the formula: In this case, of course, it is assumed that the improper integral converges. Definition. dispersion continuous random variable is called the mathematical expectation of the square of its deviation. By analogy with the variance of a discrete random variable, the following formula is used for the practical calculation of the variance: Definition. Standard deviation is called the square root of the variance. Definition. Fashion M 0 of a discrete random variable is called its most probable value. For a continuous random variable, the mode is the value of the random variable at which the distribution density has a maximum. If the distribution polygon for a discrete random variable or the distribution curve for a continuous random variable has two or more maxima, then such a distribution is called bimodal or multimodal. If a distribution has a minimum but no maximum, then it is called antimodal. Definition. Median M D of a random variable X is its value, relative to which it is equally likely to obtain a larger or smaller value of the random variable. Geometrically, the median is the abscissa of the point at which the area bounded by the distribution curve is divided in half. Note that if the distribution is unimodal, then the mode and median coincide with the mathematical expectation. Definition. Starting moment order k random variable X is called the mathematical expectation of X k. The initial moment of the first order is equal to the mathematical expectation. Definition. Central point order k random variable X is called the mathematical expectation of the value For a discrete random variable: . For a continuous random variable: . The first order central moment is always zero, and the second order central moment is equal to the dispersion. The central moment of the third order characterizes the asymmetry of the distribution. Definition.
The ratio of the central moment of the third order to the standard deviation in the third degree is called asymmetry coefficient. Definition.
To characterize the sharpness and flatness of the distribution, a quantity called kurtosis. In addition to the quantities considered, the so-called absolute moments are also used: Absolute starting moment: . Absolute central moment: . The absolute central moment of the first order is called arithmetic mean deviation. Example. For the example considered above, determine the mathematical expectation and variance of the random variable X. Example. An urn contains 6 white and 4 black balls. A ball is removed from it five times in a row, and each time the ball taken out is returned back and the balls are mixed. Taking the number of extracted white balls as a random variable X, draw up the law of distribution of this quantity, determine its mathematical expectation and variance. Because balls in each experiment are returned back and mixed, then the trials can be considered independent (the result of the previous experiment does not affect the probability of occurrence or non-occurrence of an event in another experiment). Thus, the probability of a white ball appearing in each experiment is constant and equal to Thus, as a result of five successive trials, the white ball may not appear at all, appear once, twice, three, four or five times. To draw up a distribution law, you need to find the probabilities of each of these events. 1) The white ball did not appear at all: 2) The white ball appeared once: 3) The white ball will appear twice: . By their physical nature, random variables can be deterministic and random. Discrete is a random variable whose individual values can be renumbered (the number of products, the number of parts - defective and good, etc.). A random variable is called continuous, the possible values of which fill a certain gap (deviation of the size of the manufactured part from the nominal value, measurement error, deviation of the part shape, height of microroughness, etc.). A random variable cannot be characterized by any single value. For it, it is necessary to indicate the set of possible values and the probabilistic characteristics given on this set. In the event that a random event is expressed as a number, we can talk about a random variable. Random they call the value that, as a result of the test, will take one possible value, unknown in advance and depending on random causes that cannot be taken into account in advance. Loss of some value of a random variable X this is a random event: X \u003d x i. Among random variables, discrete and continuous random variables are distinguished. Discrete random variable a random variable is called, which, as a result of the test, takes on individual values with certain probabilities. The number of possible values of a discrete random variable can be finite or infinite. Examples of a discrete random variable: recording speedometer readings or measured temperature at specific points in time. Continuous random variable a random variable is called, which, as a result of the test, takes all values from a certain numerical interval. The number of possible values of a continuous random variable is infinite. An example of a continuous random variable: measuring the speed of movement of any type of transport or temperature during a specific time interval. Any random variable has its own probability distribution law and its own probability distribution function. Before defining the distribution function, let's consider the variables that define it. Let some X is a real number and a random variable is obtained X, wherein x > X. It is required to determine the probability that the random variable X will be less than this fixed value X. The distribution function of a random variable X called a function F(x), which determines the probability that the random variable X as a result of the test will take a value less than the value of x, that is: A random variable is characterized in probability theory the law of its distribution
. This law establishes a connection between the possible values of a random variable and the probabilities of their occurrence corresponding to these values. There are two forms of describing the law of distribution of a random variable - differential and integral
. Moreover, in metrology, the differential form is mainly used - the distribution law probability density
random variable. Differential distribution law characterized probability distribution density f(x)
random variable X. Probability R hitting a random variable in the interval from x 1 before x 2 is given by the formula: Graphically, this probability is the ratio of the area under the curve f (x) in the range from x 1 to x 2 to the total area bounded by the entire distribution curve. As a rule, the area under the entire probability distribution curve is normalized to one. In this case, the distribution continuous random variable. In addition to them, there are discrete random variables that take on a number of specific values that can be numbered. Integral distribution law of a random variable is a function F(x), defined by the formula The probability that a random variable will be less than x 1 is given by the value of the function F(x) at x = x 1: Although the law of distribution of random variables is their complete probabilistic characteristic, finding this law is a rather difficult task and requires numerous measurements. Therefore, in practice, to describe the properties of a random variable, various numerical characteristics of distributions. These include moments random variables: primary and central, which are some average values. Moreover, if the values counted from the origin are averaged, then the moments are called initial, and if from the distribution center, then central. The distribution function of a random variable X is the function F(x), expressing for each x the probability that the random variable X takes the value, smaller x
Example 2.5. Given a series of distribution of a random variable Find and graphically depict its distribution function. Solution. According to the definition F(jc) = 0 for X X F(x) = 0.4 + 0.1 = 0.5 at 4 F(x) = 0.5 + 0.5 = 1 at X > 5. So (see Fig. 2.1): Distribution function properties: 1. The distribution function of a random variable is a non-negative function enclosed between zero and one: 2. The distribution function of a random variable is a non-decreasing function on the entire number axis, i.e. at X 2
>x 3. At minus infinity, the distribution function is equal to zero, at plus infinity, it is equal to one, i.e. 4. Probability of hitting a random variable X in the interval is equal to the definite integral of its probability density ranging from A before b(see Fig. 2.2), i.e. Rice. 2.2 3. The distribution function of a continuous random variable (see Fig. 2.3) can be expressed in terms of the probability density using the formula: F(x)= Jp(*)*. (2.10) 4. Improper integral in infinite limits of the probability density of a continuous random variable is equal to one: Geometric properties / and 4
probability densities mean that its plot is distribution curve - lies not below the x-axis, and the total area of the figure, limited distribution curve and x-axis, is equal to one. For a continuous random variable X expected value M(X) and variance D(X) are determined by the formulas: (if the integral converges absolutely); or (if the reduced integrals converge). Along with the numerical characteristics noted above, the concept of quantiles and percentage points is used to describe a random variable. q level quantile(or q-quantile) is such a valuex qrandom variable, at which its distribution function takes the value, equal to q, i.e. According to example 2.6 find the quantile xqj and 30% random variable point x.
Solution. By definition (2.16) F(xo t3)= 0.3, i.e. ~Y~ = 0.3, whence the quantile x 0 3 = 0.6. 30% random variable point X, or quantile Х)_о,з = xoj» is found similarly from the equation ^ = 0.7. whence *,= 1.4. ? Among the numerical characteristics of a random variable, there are initial v* and central R* k-th order moments, determined for discrete and continuous random variables by the formulas:
.
. Find the distribution density of their sum. Find the distribution of the sum of independent random variables x and h, where x has a uniform distribution on the interval, and h has an exponential distribution with parameter l. Find P 
, if x has: a) normal distribution with parameters a and s2 ; b) exponential distribution with parameter l; c) uniform distribution on the interval [-1;1]. The joint distribution of x, h is uniform squared
K = (x, y): |x| +|y|£ 2). Find Probability 
. Are x and h independent? A pair of random variables x and h is uniformly distributed inside the triangle K=. Calculate the density x and h. Are these random variables independent? Find the probability. Random variables x and h are independent and uniformly distributed on the intervals and [-1,1]. Find the probability. A two-dimensional random variable (x, h) is uniformly distributed in a square with vertices (2,0), (0,2), (-2, 0), (0,-2). Find the value of the joint distribution function at the point (1, -1). The random vector (x, h) is uniformly distributed inside a circle of radius 3 centered at the origin. Write an expression for the joint distribution density. Determine if these random variables are dependent. Calculate the probability. A pair of random variables x and h is uniformly distributed inside a trapezoid with vertices at the points (-6.0), (-3.4), (3.4), (6.0). Find the joint distribution density for this pair of random variables and the density of the components. Are x and h dependent? A random pair (x, h) is evenly distributed inside the semicircle. Find the densities x and h, investigate the question of their dependence. The joint density of two random variables x and h is 
.
Find the densities x, h. Explore the question of the dependence of x and h. A random pair (x, h) is uniformly distributed on the set . Find the densities x and h, investigate the question of their dependence. Find M(xh). Random variables x and h are independent and are distributed according to the exponential law with the parameter Find
