Continuous random variables. Random Variables Online Presentation Discrete Random Variables

DISCRETE DISTRIBUTION LAW

Plot the possible values of a random variable, and on the y-axis - the probabilities of these values.

Distribution function and its properties. Distribution function of a discrete random variable.

Distribution density and its properties

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Discrete random variables Random variables that take only values separated from each other, which can be enumerated in advance Examples: - the number of heads on three coin tosses; - the number of hits on the target with 10 shots; - the number of calls received at the ambulance station per day.

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 of a random variable can be given in the form: a table of a graph of a formula (analytically).

Calculation of the probability of realization of certain values of a random number 0.5*0.5 = 0.5 Number of heads is 2 - events: 00 - probability 0.5 *0.5 = 0.25 Sum of probabilities: 0.25 + 0.50 + 0.25 = 1

Calculation of the values of a series of distributions of a random number Problem. The shooter fires 3 shots at the target. The probability of hitting the target with each shot is 0.4. For each hit, the shooter is awarded 5 points. Construct a series of distribution of the number of scored points. Probability of events: binomial distribution Event designation: hit - 1, missed - 0 Complete group of events: 000, 100, 010, 001, 110, 101, 011, 111 k = 0, 1, 2, 3

Distribution series of a random number of scored event points number of points event probability 0.2160.4320.2880.064

Operations of addition and multiplication of random variables The sum of two random variables X and Y is a random variable, which is obtained by adding all values of a random variable X and all values of a random variable Y, the corresponding probabilities are multiplied X01 p0,20,70,1 Y123 p0,30, 50.2

Addition operations of random variables Z = = =2 0+1 =1 0+2 =2 0+3 =3 1+1 =2 1+2 =3 1+3 =4 p 0.060.10.040.210.350.140.030.050.02 02

Operations of multiplication of random variables The product of two random variables X and Y is a random variable, which is obtained by multiplying all values of the random variable X and all values of the random variable Y, the corresponding probabilities are multiplied X01 p0,20,70,1 Y123 p0,30,50, 2

Properties of the distribution function F(X) 0 F(x) 1 F(X) - non-decreasing function

Main characteristics of discrete random variables Expected value(average value) of a random variable is equal to the sum of the products of the values taken by this value by the probabilities corresponding to them: M (x) \u003d x 1 P 1 + x 2 P x n P n \u003d

Xixi PiPi x i P i (x i - M) 2 (x i - M) 2 P i 2 0.1 0.2 (2-3.6) 2 = 2.560.256 30.30.9 (3-3.6) 2 = 0.360.108 40.52 (4-3.6) 2 = 0.160.08 50.10.50.5 (5-3.6) 2 = 1.960.196 EXAMPLE: Calculate the basic numerical characteristics for the number of drug orders received per hour M( x)=3.6 D(x)=0.64
RECOMMENDED READING: Main literature: Ganicheva A.V., Kozlov V.P. Mathematics for psychologists. M.: Aspect-press, 2005, with Pavlushkov I.V. Fundamentals of higher mathematics and mathematical statistics. M., GEOTAR-Media, Zhurbenko L. Mathematics in examples and tasks. M .: Infra-M, Teaching aids: Shapiro L.A., Shilina N.G. Guide to practical exercises in medical and biological statistics Krasnoyarsk: Polikom LLC. – 2003.

Random variables are quantities that, as a result of experience, take certain values, and it is not known in advance which ones.

Designate: X,Y,Z

An example of a random variable would be:

1) X - the number of points that appears when a dice is thrown

2) Y - the number of shots before the first hit on the target

3) The height of a person, the dollar exchange rate, the player's winnings, etc.

A random variable that takes a countable set of values is called discrete.

If the set of values of r.v. Uncountable, then such a quantity is called continuous.

A random variable X is a numerical function defined on the space of elementary events Ω, which assigns to each elementary event W a number X(w), i.e. X=X(w),W

Example: Experience consists of tossing a coin 2 times. On the space of elementary events Ω(W1 ,W2 ,W3 ,W4 ) where W1 =GG, W2 =GR, W3 =RG, W4 =PP. We can consider the r.v. X is the number of appearance of the coat of arms. X is a function of

elementary event W2 : X(W1 )=2, X(W2 )=1, X(W3 )=1, X(W4 )=0 X is a discrete r.v. With values X1 =0, X2 =1, X3 =2.

For full description random variable is not enough just to know its possible values. You also need to know the probabilities of these values

RANDOM VALUE

Let X be a discrete r.v. that takes the values x1 ,

x2 ... xn ..

With some probability Pi =P(X=xi ), i=1,2,3…n…, which determines the probability that, as a result of the experiment, the r.v. X will take the value xi

Such a table is called near distribution

Since the events (X=x ),(X=x )… are incompatible and form

1 p i 1 2

full group, then i the sum1 of their probabilities is equal to

The broken line connecting the points (X1, P1), (X2, P2), ... is called

distribution polygon.







	x 1 x 2

A random variable X is discrete if there is a finite or countable set X1 , X2 ,…,Xn ,… such that P(X=xi ) = pi > 0

(i=1,2,…) and p1 +p2 +p3 +… =1

Example: There are 8 balls in an urn, of which 5 are white, the rest are black. 3 balls are drawn at random from it. Find the distribution law for the number of white balls in the sample.

Solution: Possible values of r.v. X – the number of white balls in the sample is x1 =0, x2 =1, x3 =2, x4 =3.

Their probabilities will be respectively

p(x0)

C 5 1 C 3 2

P2 =p(x=1)=

Control:

C 2 C1

P3 =p(x=2)=

C 5 3 C 3 0

P4 =p(x=2)=

C8 3

A universal way to specify the probability distribution law, suitable for both discrete and continuous random variables, is its distribution function.

The function F(x) is called the integral distribution function.

Geometrically, equality (1) can be interpreted as follows: F(x) is the probability that the r.v. X will take on the value that is depicted on the numerical axis by a point to the left of the point x, i.e. random point X will fall into the interval (∞, x)



The distribution function has the following properties:
1)F(x) is bounded, i.e. 0 F (x ) 1
2)F(x) is a non-decreasing function on R i.e. if, x 2 x 1 then
F(x2) F(x1)
3)F(x) vanishes at minus infinity and equals 1
plus infinity i.e.			F(∞)=0, F(+∞)=1
4) Probability of r.v. X in the interval is equal to the increment
its distribution function on this interval i.e.
P( a X b) F(b) F(a)

5) F(x) is left continuous i.e. Lim F(x)=F(x0 )

xx0

Using the distribution function, you can calculate

Equality (4) follows directly from the definition

6) If all x possible values x b of a random variable X

belong to the interval (a,b), then for its distribution function F(x)=0 for, F(x)=1 for

The most important characteristic of a continuous random variable is the probability distribution density.

A random variable X is called continuous if its

the distribution function is continuous and differentiable everywhere except for individual points.

The probability distribution density of a continuous r.v. X is called the derivative of its distribution function. Denoted f(x) F /

represents the average probability per unit length of the section , i.e. the average density of the probability distribution. Then


	P( x X x x)


That is, the distribution density is the limit of the ratio
probability of hitting a random variable in
interval		To the length ∆x of this gap,
	F (x x F (x) P( x X x x)
when ∆х→0		(6) equality follows

Those. the probability density is defined as a function f(x) satisfying the condition P ( x X x x ) f (x ) dx

The expression f(x)dx is called the probability element.

Distribution density properties:

1) f(x) is non-negative, i.e. f (x) 0

Methodological development is a presentation in electronic form.

This methodical development contains 26 slides with summary theoretical material to the section Random Variables. The theoretical material includes the concept of a random variable and is logically correctly divided into two parts: a discrete random variable and a continuous random variable. The topic of DSV includes the concept of DSV and methods of setting, numerical characteristics of DSV (mathematical expectation, variance, standard deviation, initial and central moments, mode, median). The main properties of the numerical characteristics of the DSW and the relationship between them are given. In the topic of CV, the above concepts are reflected in a similar way, the distribution functions of CV and the distribution density of CV are defined, the relationship between them is indicated, and the main types of distribution of CV are presented: uniform and normal distributions.

general lesson on the topic.

This development is applicable:

when studying the section Random Variables with the demonstration of individual slides for the effective assimilation of new material through visual perception,
when updating the basic knowledge of students
in preparing students for the final certification in the discipline.

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From the definition of a derivative it follows:

F(x x) F(x)

P( x X x x)

But according to formula (2), the ratio

Contents Random variables Discrete random variable (RSV) Law of distribution of SW Numerical characteristics DSW Theoretical moments of DSW A system of two DSWs Numerical characteristics of a system of two DSWs Continuous SW Distribution function of NSW Distribution density function of NSW Numerical characteristics of NSW SVR distribution curve Mode Median Uniform density distribution Normal distribution law. Laplace function

Random Variables A random variable (CV) is a variable that, as a result of an experiment, can take one or another value, and it is not known in advance which one it is before the experiment. They are divided into two types: discrete SV (DSV) and continuous SV (NSV)

Discrete Random Variable (DSV) DSV is such a variable, the number of possible trials of which is either finite or an infinite set, but necessarily countable. For example, the frequency of hits with 3 shots - X x 1 \u003d 0, x 2 \u003d 1, x 3 \u003d 2, x 4 \u003d 3 DSV will be fully described from a probabilistic point of view if it is indicated what probability each of the events has.

The distribution law of SW is a relation that establishes a relationship between the possible value of SW and the corresponding probabilities. Forms for specifying the distribution law: Table Distribution law CB X x 1 x 2 … x n P i p 1 p 2 … p n

2. Distribution polygon DSV distribution law P i X i x 1 x 2 x 3 x 4 p 1 p 2 p 3 p 4 Distribution polygon

Numerical characteristics of DSV Mathematical expectation is the sum of the products of CV values and their probabilities. Mathematical expectation is a characteristic of the average value of a random variable

Numerical characteristics of DSV Properties of mathematical expectation:

Numerical characteristics DSV 2. The variance of DSVH is the mathematical expectation of the square of the deviation of a random variable from the mathematical expectation. Dispersion characterizes the measure of dispersion of SW values from the mathematical expectation When solving problems, it is convenient to calculate the dispersion using the formula: - Standard deviation

Numerical characteristics of DSW Properties of dispersion:

Theoretical moments of the DSW The initial moment of order k SVR is the mathematical ratio Х k

System of two SVs A system of two SVs (Х Y) can be represented by a random point on the plane. The event consisting in the hit of a random point (X Y) in the area D is denoted by (X, Y) ∩ D

A system of two DSWs A table specifying the distribution law for a system of two DSWs Y X y 1 y 2 y 3 … y n x 1 p 11 p 12 p 13 … p 1n x 2 p 21 p 22 p 23 … p 2n x 3 p 31 p 32 p 33 … p 3n … … … … … … x m p m1 p m2 p m3 … p mn

Numerical characteristics of a system of two DSWs Mathematical expectation and variance of a system of two DSWs by definition When solving problems, it is convenient to apply the formula

Continuous SW NSW is such a quantity, the possible values of which continuously fill a certain interval (finite or infinite). The number of all possible NSV values is infinite. Example: Random deviation in range of the point of impact of the projectile from the target.

The distribution function of the CVW The distribution function is called F(x) , which determines for each value x the probability that the CVH will take a value less than x, i.e. according to the definition F(x)=P(X

Distribution function of NSW Properties of the distribution function: if, then corollary: If all possible values x of SVR belong to the interval (a;b) , then for a=b F(x)=0 Corollary: 1. 2. 3. The distribution function is left-continuous

NSV distribution density function The probability distribution density function is the first derivative of the function F(x) f(x)=F`(x). f(x) is called a differential function. The probability that the CVSH will take values belonging to the interval (a;b) calculated by the formula Knowing the distribution density, you can find the distribution function Properties: , in particular, if all possible values of CB belong to (a;b) , then 1. 2.

Numerical characteristics of NSV The mathematical expectation of NSVH, all possible values of which belong to the interval (a;b), is determined by the equality: The variance of NSWH, all possible values of which belong to the interval (a;b), is determined by the equality:

Numerical characteristics of the NSV The standard deviation is determined in the same way as for the DSV: The initial moment of the k-th order of the NSV is determined by the equality:

Numerical characteristics of the NSV The central moment of the kth order of the NSVH, all possible values of which belong to the interval (a:b), is determined by the equality:

Numerical characteristics of NSV If all possible values of NSVH belong to the entire numerical axis OX, then in all the above formulas the definite integral is replaced by an improper integral with infinite lower and upper limits

Distribution curve of SVR Y X M 0 a b Graph of the function f(x) is called the distribution curve distribution curve Geometrically, the probability of SVR falling into the interval (a; b) is equal to the area curvilinear trapezoid, bounded by the distribution curve by the OX axis and the straight lines x=a and x=b

Mode The DSWR mode is its most probable value. The NSWH mode is its value M 0 , at which the distribution density is maximum. To find the NSW mode, it is necessary to find the maximum of the function using the first or second derivative. M 0 \u003d 2, because 0.1 0.3 Geometrically, the mode is the abscissa of that point of the curve or distribution polygon, the ordinate of which is maximum X 1 2 3 P 0.1 0.6 0.3 Y X M 0 a b

Median The median of the NSVR is its value M e, for which it is equally likely that the random variable will be greater or less than M e, i.e. P(x M e)=0.5 An ordinate drawn to a point with an abscissa equal to M e bisects the area bounded by the distribution curve or polygon. If the straight line x=a is the axis of symmetry of the distribution curve y=f(x), then M 0 =M e = M(X)= a

Uniform density distribution Uniform is the distribution of such SWs, all of whose values lie on a certain segment (a;b) and have a constant probability density on this segment Y X a b h Mathematical expectation, variance, standard deviation of a uniformly distributed SW:

Normal distribution law. Laplace function The normal distribution law is characterized by density. The distribution curve is symmetrical with respect to the straight line x=a . The maximum ordinate at x=a is Y X x=a Gaussian curve, normal curve The abscissa axis is the asymptote of the curve y=f(x) Ф (x) - Laplace function