The concept of a random variable. Distribution law of a random variable
Random variables (abbreviated: r.v.) are denoted by capital Latin letters X, Y, Z,...(or lowercase Greek letters ξ (xi), η (this), θ (theta), ψ (psi), etc.), and the values taken by them, respectively, in small letters x 1 , x 2 ,…, 1 , at 2 , 3 …
Examples With. in. can serve: 1) X- the number of points that appear when throwing a dice; 2) Y - the number of shots before the first hit on the target; 3) Z- uptime of the device, etc. (human height, dollar exchange rate, number of defective parts in a batch, air temperature, player's gain, coordinate of a point if it is randomly selected by , company's profit, ...).
Random variable XΏ w
X(w), i.e. X= X(w), wО Ώ (or X=f(w)) (31)
Example1. The experience consists in tossing a coin 2 times. On the PES Ώ=( w 1 , w 2 , w 3 , w 4 ), where w 1 = GG, w 2 = GR, w 3 = RG, w 4 = RR, you can consider with. in. X- the number of appearances of the coat of arms. S. v. X is a function of the elementary event w i :X( w 1 ) = 2, X( w 2 ) = 1, X( w 3 ) = 1, X( w 4 )= 0; X- d.s. in. with values x 1 = 0,x2 =1 , x 3 = 2.
X(w) S P(A) = P(X< X).
X- d.s. in.,
x 1 , x 2 , x 3 ,…,x n ,…
p i , where i = 1,2,3, ...,n,… .
distribution law d.s. in. p i =P(X=x i}, i=1,2,3,...,n,...,
With. in. X x i . :
| X | x 1 | x2 | …. | x n | … |
| P | p1 | p2 | …. | p n | … |
Since the events (X= x 1 ), (X= x 2 ),…, (X= x n ), i.e. .
(x 1 , p1 ), (x 2 , p 2),…, (x n , p n) are called polygon(or polygon) distribution(see fig. 17).
Random value X is discrete, if there is a finite or countable set of numbers x 1 , x2 , ..., x n such that P(X = x i ) = p i > 0 (i = 1,2,...) p 1 + p2 + p 3 +…= 1 (32)
sum d.s. in. X, which takes the values x i with probabilities p i = Р(Х = x i ), i = 1,2,3,...,n, and d.s. in. Y, taking values y j with probabilities p i = Р(Y = y j ), j = 1,2,3,...,m, is called a d.s. in. Z = X + Y , taking the values z ij = x i + y j with probabilities p ij = Р( Х = x i ,Y = y j ), for all specified values i and j. If some sums x i + y j coincide, the corresponding probabilities are added.
difference d.s. in. X, which takes the values x i with probabilities p i = Р(Х = x i ), i = 1,2,3,...,n, and d.s. in. Y, taking values y j with probabilities p i = P(Y = y j ), j = 1,2,3,...,m, is called a d.s. in. Z = X - Y, taking the values z ij = x i – y j with probabilities p ij = Р( Х = x i ,Y = y j ), for all specified values i and j. If some differences x i – y j coincide, the corresponding probabilities are added.
work d.s. in. X, which takes the values x i with probabilities p i = Р(Х = x i ), i = 1,2,3,...,n, and d.s. in. Y, taking values y j with probabilities p i = P(Y = y j ), j = 1,2,3,...,m, is called a d.s. in. Z = X × Y, taking the values z ij = x i × y j with probabilities p ij = Р( Х = x i ,Y = y j ), for all specified values i and j. If some products x i × y j coincide, the corresponding probabilities are added.
d.s. in. сХ, с x i р i = Р(Х = x i ).
X and Y events (X = x i ) = А i and (Y = y j ) = В j are independent for any i= 1,2,...,n; j = l,2,...,m, i.e.,
P(X = x i ;Y = y j ) =P(X = x i ) ×P (Y = y j ) (33)
Example 2 There are 8 balls in an urn, 5 of which are white and 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.
Random value is a quantity that, as a result of the experiment, takes on a previously unknown value.
The number of students attending the lecture.
The number of houses commissioned in the current month.
Ambient temperature.
The weight of a fragment of an exploding projectile.
Random variables are divided into discrete and continuous.
Discrete (discontinuous) called a random variable that takes on separate, isolated from each other values with certain probabilities.
The number of possible values of a discrete random variable can be finite or countable.
continuous is called 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.
In the given examples: 1 and 2 are discrete random variables, 3 and 4 are continuous random variables.
In the future, instead of the words "random variable" we will often use the abbreviation c. in.
As a rule, random variables will be denoted by capital letters, and their possible values by small letters.
In the set-theoretic interpretation of the basic concepts of probability theory, a random variable X is a function of an elementary event: X =φ(ω), where ω is an elementary event belonging to the space Ω (ω Ω). In this case, the set Ξ of possible values of c. in. X consists of all the values that the function φ(ω) takes.
The law of distribution of a random variable Any rule (table, function) is called that allows you to find the probabilities of all kinds of events associated with a random variable (for example, the probability that it will take some value or fall into some interval).
Forms of setting the laws of distribution of random variables. Distribution range.
This is a table in the top line of which all possible values of the random variable X are listed in ascending order: x 1, x 2, ..., x n, and in the bottom - the probabilities of these values: p 1, p 2, ..., p n, where p i \u003d P (X \u003d x i).
Since the events (X \u003d x 1), (X \u003d x 2), ... are incompatible and form a complete group, the sum of all probabilities in the bottom line of the distribution series is equal to one
The distribution series is used to set the distribution law for only discrete random variables.
Distribution polygon
The graphic representation of a distribution series is called a distribution polygon. It is built like this: for each possible value c. in. the perpendicular to the x-axis is restored, on which the probability of a given value c is plotted. in. The obtained points for clarity (and only for clarity!) are connected by line segments.
The cumulative distribution function (or just the distribution function).
This is a function that, for each value of the argument x, is numerically equal to the probability that the random variable will be less than the value of the argument x.
The distribution function is denoted by F(x): F(x) = P (X x).
Now we can give a more precise definition of a continuous random variable: a random variable is called continuous if its distribution function is a continuous, piecewise differentiable function with a continuous derivative.
The distribution function is the most versatile form of setting c. in., which can be used to set the laws of distribution of both discrete and continuous s. in.
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Graphically, the law of distribution of a discrete quantity is given in the form of a so-called distribution polygon.
The graphic representation of the distribution series (see Fig. 5) is called the distribution polygon.
To characterize the distribution law of a discontinuous random variable, a series (table) and a distribution polygon are often used.
For its image in a rectangular coordinate system, points are built (Y Pi) (x - i Pa) and connected by line segments. The distribution polygon gives an approximate visual representation of the nature of the distribution of a random variable.
For clarity, the distribution law of a discrete random variable can also be depicted graphically, for which points (x /, p) are built in a rectangular coordinate system, and then they are connected by line segments. The resulting figure is called the distribution polygon.
M (xn; pn) (ls - - possible values of Xt pi - corresponding probabilities) and connect them with line segments. The resulting figure is called the distribution polygon.
Consider the probability distribution of the sum of points on dice. The figures below show the distribution polygons for the case of one, two and three bones.
In this case, instead of a random distribution polygon, a distribution density function is constructed, which is called the differential distribution function and is a differential distribution law. In probability theory, the distribution density of a random variable x (x Xr) is understood as the limit of the ratio of the probability that x falls into the interval (x, x - - Ax) to Ax, when Al; tends to zero. In addition to the differential function, to characterize the distribution of a random variable, the integral distribution function is used, which is often called simply the distribution function or the integral distribution law.
With such a construction, the relative frequencies of falling into the intervals will be equal to the areas of the corresponding columns of the histogram, just as the probabilities are equal to the areas of the corresponding curvilinear trapezoids. y Sometimes, for clarity of comparison, a distribution polygon is built, connecting in series the midpoints of the upper bases of the histogram bars.
By giving m different values from 0 to z, the probabilities PQ, P RF - Pp are obtained, which are plotted on the graph. Given r; i11, construct a polygon of the probability distribution.
The distribution law of a discrete random variable is any correspondence between its possible values and their probabilities. The law can be specified tabularly (distribution series), graphically (distribution polygon, etc.) and analytically.
Finding the distribution curve, in other words, establishing the distribution of the random variable itself, makes it possible to more deeply investigate the phenomenon, which is far from being fully expressed by this particular distribution series. By presenting on the drawing both the found leveling distribution curve and the distribution polygon constructed on the basis of a partial population, the researcher can clearly see the characteristic features inherent in the phenomenon under study. Due to this, statistical analysis detains the researcher's attention on the deviations of the observed data from some regular change in the phenomenon, and the researcher faces the task of finding out the causes of these deviations.
Then, abscissas (on a scale) are drawn from the middle of the intervals, corresponding to the number of months with flow in this interval. The ends of these abscissas are connected and, thus, a polygon, or distribution polygon, is obtained.
Points that give a graphical representation of the distribution law of a discrete random variable on the coordinate plane of the value of the value - the probability of values, are usually connected by line segments and the resulting geometric figure is called the distribution polygon. On fig. 3 in Table 46 (as well as in Figures 4 and 5) just shows the distribution polygons.
Discrete called a random variable that can take on separate, isolated values with certain probabilities.
EXAMPLE 1. The number of occurrences of the coat of arms in three coin tosses. Possible values: 0, 1, 2, 3, their probabilities are equal respectively:
P(0) = ; P(1) = ; P(2) = ; P(3) = .
EXAMPLE 2. The number of failed elements in a device consisting of five elements. Possible values: 0, 1, 2, 3, 4, 5; their probabilities depend on the reliability of each of the elements.
Discrete random variable X can be given by a distribution series or a distribution function (an integral distribution law).
Near distribution is the set of all possible values Xi and their corresponding probabilities Ri = P(X = xi), it can be given as a table:
| x i | x n |
|||
| p i | p n |
At the same time, the probabilities Ri satisfy the condition
Ri= 1 because 
where is the number of possible values n may be finite or infinite.
Graphical representation of a distribution series called the distribution polygon . To construct it, the possible values of the random variable ( Xi) are plotted along the x-axis, and the probabilities Ri- along the y-axis; points BUTi with coordinates ( Xi ,pi) are connected by broken lines.
distribution function random variable X called a function F(X), whose value is at the point X is equal to the probability that the random variable X will be less than this value X, that is
F(x) = P(X< х).
Function F(X) for discrete random variable calculated by the formula
F(X) = Ri , (1.10.1)
where the summation is over all values i, for which Xi< х.
EXAMPLE 3. From a batch containing 100 items, among which there are 10 defective items, five items are randomly selected to check their quality. Construct a series of distributions of a random number X defective products contained in the sample.
Solution. Since the number of defective products in the sample can be any integer in the range from 0 to 5 inclusive, the possible values Xi random variable X are equal:
x 1 = 0, x 2 = 1, x 3 = 2, x 4 = 3, x 5 = 4, x 6 = 5.
Probability R(X = k) that in the sample will be exactly k(k = 0, 1, 2, 3, 4, 5) defective products, equal to
P (X \u003d k) \u003d.
As a result of calculations using this formula with an accuracy of 0.001, we obtain:
R 1 = P(X = 0) @ 0,583;R 2 = P(X = 1) @ 0,340;R 3 = P(X = 2) @ 0,070;
R 4 = P(X = 3) @ 0,007;R 5 = P(X= 4) @ 0;R 6 = P(X = 5) @ 0.
Using equality to check Rk=1, we make sure that the calculations and rounding are done correctly (see table).
| x i | ||||||
| p i |
EXAMPLE 4. Given a series of distribution of a random variable X :
| x i | |||||
| p i |
Find the probability distribution function F(X) of this random variable and construct it.
Solution. If a X£10 then F(X)= P(X<X) = 0;
if 10<X£20 then F(X)= P(X<X) = 0,2 ;
if 20<X£30 then F(X)= P(X<X) = 0,2 + 0,3 = 0,5 ;
if 30<X£40 then F(X)= P(X<X) = 0,2 + 0,3 + 0,35 = 0,85 ;
if 40<X£50 then F(X)= P(X<X) = 0,2 + 0,3 + 0,35 + 0,1=0,95 ;
if X> 50 , then F(X)= P(X<X) = 0,2 + 0,3 + 0,35 + 0,1 + 0,05 = 1.
