What is the distribution function of a random variable. The probability distribution function of a random variable and its properties
We have established that the distribution series completely characterizes a discrete random variable. However, this characteristic is not universal. It exists only for discrete quantities. For a continuous quantity, a distribution series cannot be constructed. Indeed, continuous random value has an infinite number of possible values that completely fill a certain gap. It is impossible to compile a table in which all possible values of this quantity would be listed. Therefore, for a continuous random variable, there is no distribution series in the sense in which it exists for discrete quantity. However, different ranges of possible values of a random variable are not equally probable, and there is still a "probability distribution" for a continuous variable, although not in the same sense as for a discrete one.
To quantify this probability distribution, it is convenient to use the non-probability of the event R(X= X), consisting in the fact that the random variable will take a certain value X, and the probability of an event R(X<X), consisting in the fact that the random variable will take a value less than X. Obviously, the probability of this event depends on X, i.e. is some function of X.
Definition. distribution function random variable X called a function F(x) expressing for each value X the probability that the random variable X takes on a value less than X:
| F(x) = P(X < x). | (4.2) |
The distribution function is also called cumulative distribution function or integral distribution law .
The distribution function is the most universal characteristic of a random variable. It exists for all random variables: both discrete and continuous. The distribution function completely characterizes a random variable from a probabilistic point of view, i.e. is a form of distribution law.
The distribution function allows a simple geometric interpretation. Consider a random variable X on axle Oh(Fig. 4.2), which, as a result of the experiment, can take one position or another. Let a point be selected on the axis, which has the value X. Then, as a result of the experiment, the random variable X may be to the left or right of the point X. Obviously, the probability that the random variable X will be to the left of the point X, will depend on the position of the point X, i.e. be a function of the argument X.
For a discrete random variable X, which can take the values X 1 , X 2 , …, x n, the distribution function has the form
Find and graphically depict its distribution function.
Solution. We will set different values X and find for them F(x) = = P(X < x).
1. If X≤ 0, then F(x) = P(X < X) = 0.
2. If 0< X≤ 1, then F(x) = P(X < X) = P(X = 0) = 0,08.
3. If 1< X≤ 2, then F(x) = P(X < X) = P(X = 0) + P(X = 1) = 0,08 + 0,44 = 0,52.
4. If X> 2, then F(x) = P(X < X) = P(X = 0) + P(X = 1) + P(X = 2) = 0,08 + 0,44 + + 0,48 = 1.
Let's write the distribution function.
Let's depict the distribution function graphically (Fig. 4.3). Note that when approaching the discontinuity points from the left, the function retains its value (such a function is said to be continuous from the left). These points are highlighted on the graph. ◄
This example leads to the assertion that the distribution function of any discrete random variable is a discontinuous step function whose jumps occur at points corresponding to the possible values of the random variable and are equal to the probabilities of these values.
Consider general properties distribution functions.
1. The distribution function of a random variable is a non-negative function between zero and one:
3. At minus infinity the distribution function is equal to zero, at plus infinity it is equal to one, i.e.
Example 4.3. Distribution function of a random variable X looks like:
Find the probability that the random variable X takes a value in the interval and having zero probability.
However, the concept of an event that has a non-zero probability, but which consists of events with zero probability, is no more paradoxical than the idea of a segment having a certain length, while not a single point of the segment has a non-zero length. A segment consists of such points, but its length is not equal to the sum of their lengths.
The following corollary follows from this property.
Consequence. If X is a continuous random variable, then the probability that this variable falls into the interval (x 1, x 2) does not depend on whether this interval is open or closed:
P(x 1 < X < x 2) = P(x 1 ≤ X < x 2) = P(x 1 < X ≤ x 2) = P(x 1 ≤ X ≤ x 2).
The distribution function is the most general form setting the distribution law. It is used to specify both discrete and continuous random variables. It is usually referred to as . distribution function determines the probability that a random variable takes values less than a fixed real number, i.e. 
. The distribution function completely characterizes a random variable from a probabilistic point of view. It is also called the integral distribution function.
The geometric interpretation of the distribution function is very simple. If a random variable is considered as a random point of the axis (Fig. 6), which, as a result of the test, can take one or another position on this axis, then the distribution function is the probability that the random point, as a result of the test, will fall to the left of the point.
For a discrete random variable , which can take the values,, … ,, the distribution function has the form
,
where the inequality under the sum sign means that the summation extends to all those values that are smaller in magnitude. It follows from this formula that the distribution function of a discrete random variable is discontinuous and increases in jumps when passing through the points,, …,, and the jump is equal to the probability of the corresponding value (Fig. 7). The sum of all jumps in the distribution function is equal to one.
A continuous random variable has a continuous distribution function, the graph of this function has the form of a smooth curve (Fig. 8).
Rice. 7. Fig. eight.
Consider the general properties of distribution functions.
Property 1. The distribution function is a non-negative function enclosed between zero and one:
The validity of this property follows from the fact that the distribution function is defined as the probability of a random event consisting in that.
Property 2. The probability of a random variable falling into an interval is equal to the difference between the values of the distribution function at the ends of this interval, i.e.
It follows that the probability of any single value of a continuous random variable is zero.
Property 3. The distribution function of a random variable is a non-decreasing function, i.e., for .
Property 4. At minus infinity, the distribution function is zero, and at plus infinity, the distribution function is equal to unity, i.e.,.
Example 1 The distribution function of a continuous random variable is given by the expression
Find the coefficient and build a graph. Determine the probability that a random variable as a result of the experiment will take a value on the interval.
Solution. Since the distribution function of a continuous random variable is continuous, we get: 
. From here. The graph of the function is shown in Fig. 9.
Based on the second property of the distribution function, we have:
.
4. Probability distribution density and its properties.
The distribution function of a continuous random variable is its probabilistic characteristic. But it has a drawback, which consists in the fact that it is difficult to judge the nature of the distribution of a random variable in a small neighborhood of one or another point of the numerical axis. A more visual representation of the nature of the distribution of a continuous random variable is given by a function called the probability distribution density or differential distribution function of a random variable.
Distribution density is equal to the derivative of the distribution function, i.e.
.
The meaning of the distribution density is that it indicates how often a random variable appears in a certain neighborhood of a point when experiments are repeated. The curve representing the distribution density of a random variable is called distribution curve.
Consider the properties of the distribution density.
Property 1. The distribution density is non-negative, i.e.
Property 2. The distribution function of a random variable is equal to the integral of the density in the interval from to, i.e.
To find the distribution functions of random variables and their variables, it is necessary to study all the features of this field of knowledge. There are several various methods to find the values under consideration, including variable change and torque generation. Distribution is a concept based on such elements as dispersion, variations. However, they characterize only the degree of scattering range.
The more important functions of random variables are those that are related and independent, and equally distributed. For example, if X1 is the weight of a randomly selected individual from a population of males, X2 is the weight of another, ..., and Xn is the weight of another person from a male population, then we need to know how random function X is distributed. In this case, the classical theorem called the central limit theorem applies. It allows us to show that for large n the function follows standard distributions.
Functions of one random variable
The central limit theorem is designed to approximate discrete values in question, such as binomial and Poisson. Distribution functions of random variables are considered, first of all, on simple values of one variable. For example, if X is a continuous random variable having its own probability distribution. In this case, we are exploring how to find the Y density function using two different approaches, namely the distribution function method and variable change. First, only one-to-one values are considered. Then you need to modify the technique of changing the variable to find its probability. Finally, one needs to learn how the cumulative distribution can help model random numbers that follow certain sequential patterns.
Method of distribution of considered values
The method of the probability distribution function of a random variable is applicable in order to find its density. When using this method, a cumulative value is calculated. Then, by differentiating it, you can get the probability density. Now that we have the distribution function method, we can look at a few more examples. Let X be a continuous random variable with a certain probability density.
What is the probability density function of x2? If you look at or graph the function (top and right) y \u003d x2, you can note that it is an increasing X and 0 In the last example, great care was used to index the cumulative functions and the probability density with either X or Y to indicate which random variable they belonged to. For example, when finding the cumulative distribution function Y, we got X. If you need to find a random variable X and its density, then you just need to differentiate it. Let X be a continuous random variable given by a distribution function with a common denominator f(x). In this case, if you put the value of y in X = v (Y), then you get the value of x, for example v (y). Now, we need to get the distribution function of a continuous random variable Y. Where the first and second equality takes place from the definition of cumulative Y. The third equality holds because the part of the function for which u (X) ≤ y is also true that X ≤ v (Y ). And the latter is done to determine the probability in a continuous random variable X. Now we need to take the derivative of FY (y), the cumulative distribution function of Y, to get the probability density of Y. Let X be a continuous random variable with common f(x) defined over c1 To address this issue, quantitative data can be collected and an empirical cumulative distribution function can be used. With this information and appealing to it, you need to combine means samples, standard deviations, media data, and so on. Similarly, even a fairly simple probabilistic model can have a huge number of results. For example, if you flip a coin 332 times. Then the number of results obtained from flips is greater than that of google (10100) - a number, but not less than 100 quintillion times higher than elementary particles in the known universe. Not interested in an analysis that gives an answer to every possible outcome. A simpler concept would be needed, such as the number of heads, or the longest stroke of the tails. To focus on issues of interest, a specific result is accepted. The definition in this case is as follows: a random variable is a real function with a probability space. The range S of a random variable is sometimes called the state space. Thus, if X is the value in question, then so N = X2, exp ↵X, X2 + 1, tan2 X, bXc, and so on. The last of these, rounding X to the nearest whole number, is called the floor function. Once the distribution function of interest for the random variable x is determined, the question usually becomes: "What are the chances that X falls into some subset of the values of B?". For example, B = (odd numbers), B = (greater than 1), or B = (between 2 and 7) to indicate those results that have X, the value of the random variable, in subset A. So in the above example, you can describe the events as follows. (X is an odd number), (X is greater than 1) = (X > 1), (X is between 2 and 7) = (2 Thus, it is possible to calculate the probability that the distribution function of a random variable x will take values in the interval by subtracting. Consideration needs to be given to including or excluding endpoints. We will call a random variable discrete if it has a finite or countably infinite state space. Thus, X is the number of heads on three independent flips of a biased coin that goes up with probability p. We need to find the cumulative distribution function of a discrete random variable FX for X. Let X be the number of peaks in a collection of three cards. Then Y = X3 via FX. FX starts at 0, ends at 1, and does not decrease as x values increase. The cumulative FX distribution function of a discrete random variable X is constant, except for jumps. When jumping the FX is continuous. It is possible to prove the statement about the correct continuity of the distribution function from the probability property using the definition. It sounds like this: a constant random variable has a cumulative FX that is differentiable. To show how this can happen, we can give an example: a target with a unit radius. Presumably. the dart is evenly distributed over the specified area. For some λ> 0. Thus, the distribution functions of continuous random variables increase smoothly. FX has the properties of a distribution function. A man waits at a bus stop until the bus arrives. Having decided for himself that he will refuse when the wait reaches 20 minutes. Here it is necessary to find the cumulative distribution function for T. The time when a person will still be at the bus station or will not leave. Despite the fact that the cumulative distribution function is defined for each random variable. All the same, other characteristics will be used quite often: the mass for a discrete variable and the distribution density function of a random variable. Usually the value is output through one of these two values. These values are considered by the following properties, which are of a general (mass) character. The first is based on the fact that the probabilities are not negative. The second follows from the observation that the set for all x=2S, the state space for X, forms a partition of the probabilistic freedom of X. Example: tossing a biased coin whose outcomes are independent. You can continue to perform certain actions until you get a throw of heads. Let X denote a random variable that gives the number of tails in front of the first head. And p denotes the probability in any given action. So, the mass probability function has the following characteristic features. Because the terms form a numerical sequence, X is called a geometric random variable. Geometric scheme c, cr, cr2,. , crn has a sum. And, therefore, sn has a limit as n 1. In this case, the infinite sum is the limit. The mass function above forms a geometric sequence with a ratio. Therefore, natural numbers a and b. The difference in values in the distribution function is equal to the value of the mass function. The density values under consideration have the following definition: X is a random variable whose distribution FX has a derivative. FX satisfying Z xFX (x) = fX (t) dt-1 is called the probability density function. And X is called a continuous random variable. In the fundamental theorem of calculus, the density function is the derivative of the distribution. You can calculate probabilities by calculating definite integrals. Because data are collected from multiple observations, more than one random variable at a time must be considered in order to model the experimental procedures. Therefore, the set of these values and their joint distribution for the two variables X1 and X2 means viewing events. For discrete random variables, joint probabilistic mass functions are defined. For continuous ones, fX1, X2 are considered, where the joint probability density is satisfied. Two random variables X1 and X2 are independent if any two events associated with them are the same. In words, the probability that two events (X1 2 B1) and (X2 2 B2) occur at the same time, y, is equal to the product of the variables above, that each of them occurs individually. For independent discrete random variables, there is a joint probabilistic mass function, which is the product of the limiting ion volume. For continuous random variables that are independent, the joint probability density function is the product of the marginal density values. Finally, n independent observations x1, x2, are considered. , xn arising from an unknown density or mass function f. For example, an unknown parameter in functions for an exponential random variable describing the waiting time for a bus. The main goal of this theoretical field is to provide the tools needed to develop inferential procedures based on sound principles of statistical science. Thus, one very important use case for software is the ability to generate pseudo-data to mimic actual information. This makes it possible to test and improve analysis methods before having to use them in real databases. This is required in order to explore the properties of the data through modeling. For many commonly used families of random variables, R provides commands for generating them. For other circumstances, methods for modeling a sequence of independent random variables that have a common distribution will be needed. Discrete Random Variables and Sample Command. The sample command is used to create simple and stratified random samples. As a result, if a sequence x is input, sample(x, 40) selects 40 records from x such that all choices of size 40 have the same probability. This uses the default R command for fetch without replacement. Can also be used to model discrete random variables. To do this, you need to provide a state space in the vector x and the mass function f. A call to replace = TRUE indicates that sampling occurs with replacement. Then, to give a sample of n independent random variables having a common mass function f, the sample (x, n, replace = TRUE, prob = f) is used. It is determined that 1 is the smallest value represented, and 4 is the largest of all. If the command prob = f is omitted, then the sample will sample uniformly from the values in vector x. You can check the simulation against the mass function that generated the data by looking at the double equals sign, ==. And recalculating the observations that take every possible value for x. You can make a table. Repeat this for 1000 and compare the simulation with the corresponding mass function. First, simulate homogeneous distribution functions of random variables u1, u2,. , un on the interval . About 10% of the numbers should be within . This corresponds to 10% simulations on the interval for a random variable with the FX distribution function shown. Similarly, about 10% of the random numbers should be in the interval . This corresponds to 10% simulations on the random variable interval with the distribution function FX. These values on the x axis can be obtained by taking the inverse from FX. If X is a continuous random variable with density fX positive everywhere in its domain, then the distribution function is strictly increasing. In this case, FX has an inverse FX-1 function known as the quantile function. FX (x) u only when x FX-1 (u). The probability transformation follows from the analysis of the random variable U = FX(X). FX has a range from 0 to 1. It cannot take values below 0 or above 1. For values of u between 0 and 1. If U can be modeled, then it is necessary to simulate a random variable with FX distribution via a quantile function. Take the derivative to see that the density u varies within 1. Since the random variable U has a constant density over the interval of its possible values, it is called uniform on the interval. It is modeled in R with the runif command. The identity is called a probabilistic transformation. You can see how it works in the dart board example. X between 0 and 1, the distribution function u = FX(x) = x2, and hence the quantile function x = FX-1(u). It is possible to model independent observations of the distance from the center of the dart panel, while generating uniform random variables U1, U2,. , Un. The distribution function and the empirical function are based on 100 simulations of the distribution of a dart board. For an exponential random variable, presumably u = FX (x) = 1 - exp (- x), and hence x = - 1 ln (1 - u). Sometimes logic consists of equivalent statements. In this case, you need to concatenate the two parts of the argument. The intersection identity is similar for all 2 (S i i) S, instead of some value. The union Ci is equal to the state space S and each pair is mutually exclusive. Since Bi - is divided into three axioms. Each check is based on the corresponding probability P. For any subset. Using an identity to make sure the answer doesn't depend on whether the interval endpoints are included. For each outcome in all events, the second property of the continuity of probabilities is ultimately used, which is considered axiomatic. The law of distribution of the function of a random variable here shows that each has its own solution and answer. Definitions of the Distribution Function of a random variable and the Probability Density of a continuous random variable are given. These concepts are actively used in articles about site statistics. Examples of calculating the Distribution Function and Probability Density using MS EXCEL functions are considered..
Let us introduce the basic concepts of statistics, without which it is impossible to explain more complex concepts. Let us have population(population) of N objects, each of which has a certain value of some numerical characteristic X. An example of a general population (GS) is a set of weights of the same type of parts that are produced by a machine. Since in mathematical statistics, any conclusion is made only on the basis of the characteristic X (abstracting from the objects themselves), then from this point of view population represents N numbers, among which, in the general case, there may be the same. In our example, the WB is simply a numeric array of part weight values. X is the weight of one of the parts. If from a given GS we randomly select one object with characteristic X, then the value of X is random variable. By definition, any random value It has distribution function, which is usually denoted by F(x). distribution function probabilities random variable X is the function F(x), the value of which at the point x is equal to the probability of the event X F(x) = P(X Let's explain on the example of our machine. Although it is assumed that our machine produces only one type of part, it is obvious that the weight of the parts produced will vary slightly from each other. This is possible due to the fact that different materials could be used in the manufacture, and the processing conditions could also vary slightly, etc. Let the heaviest part produced by the machine weigh 200 g, and the lightest - 190 g. The probability that by chance the selected part X will weigh less than 200 g is 1. The probability that it will weigh less than 190 g is 0. Intermediate values are determined by the form of the Distribution Function. For example, if the process is set to produce parts weighing 195 g, then it is reasonable to assume that the probability of choosing a part lighter than 195 g is 0.5. Typical Graph Distribution functions for a continuous random variable is shown in the picture below (purple curve, see example file): MS EXCEL Help distribution function called integral distribution function (Cumulativedistributionfunction,
CDF). Here are some properties Distribution functions: Recall that distribution density is a derivative of distribution functions, i.e. "speed" of its change: p(x)=(F(x2)-F(x1))/Dx with Dx tending to 0, where Dx=x2-x1. Those. the fact that distribution density>1 means only that the distribution function grows fast enough (this is obvious in the example ). Note: The area entirely contained under the entire curve representing distribution density, is equal to 1. Note: Recall that the distribution function F(x) is called in MS EXCEL functions cumulative distribution function. This term appears in function parameters, such as NORM.DIST(x; mean; standard deviation; integral). If the MS EXCEL function should return distribution function, then the parameter integral, d.b. set to TRUE. If you need to calculate probability density, then the parameter integral, d.b. FALSE. Note: For discrete distribution the probability of a random variable taking on a certain value is also often called a probability mass function (pmf). MS EXCEL Help probability density can even call it a "probability measure function" (see the BINOM.DIST() function). It is clear that in order to calculate probability density for a certain value of a random variable, you need to know its distribution. Let's find probability density for N(0;1) at x=2. To do this, you need to write the formula =NORM.ST.DIST(2,FALSE)=0.054 or =NORM.DIST(2,0,1,FALSE). Recall that probability that continuous random variable will take a specific value of x equal to 0. For continuous random variable X can only calculate the probability of the event that X will take the value contained in the interval (a; b). 1) Find the probability that the random variable distributed over (see the picture above) took a positive value. According to property Distribution functions the probability is F(+∞)-F(0)=1-0.5=0.5. NORM.ST.DIST(9,999E+307;TRUE) - NORM.ST.DIST(0,TRUE) =1-0,5. 2) Find the probability that a random variable distributed over , took a negative value. According to the definition distribution functions, the probability is F(0)=0.5. In MS EXCEL, to find this probability, use the formula =NORM.ST.DIST(0,TRUE) =0,5. 3) Find the probability that a random variable distributed over standard normal distribution, will take the value contained in the interval (0; 1). The probability is F(1)-F(0), i.e. from the probability of choosing X from the interval (-∞;1) you need to subtract the probability of choosing X from the interval (-∞;0). In MS EXCEL use the formula =NORM.ST.DIST(1,TRUE) - NORM.ST.DIST(0,TRUE). All calculations above refer to a random variable distributed over standard normal law N(0;1). It is clear that the probability values depend on the specific distribution. In the article, find the point for which F (x) = 0.5, and then find the abscissa of this point. Point abscissa =0, i.e. the probability that the random variable X takes the value<0, равна 0,5. In MS EXCEL, use the formula =NORM.ST.INV(0.5) =0. Unambiguously calculate the value random variable allows the monotonicity property distribution functions. Inverse distribution function calculates , which are used, for example, when . Those. in our case, the number 0 is the 0.5-quantile normal distribution. In the example file, you can calculate another quantile this distribution. For example, the 0.8 quantile is 0.84. In English literature inverse distribution function often referred to as Percent Point Function (PPF). Note: When calculating quantiles in MS EXCEL, the following functions are used: NORM.ST.OBR() , LOGNORM.OBR() , XI2.OBR(), GAMMA.OBR(), etc. You can read more about the distributions presented in MS EXCEL in the article. The content of the article
DISTRIBUTION FUNCTION is the probability density of the distribution of particles of a macroscopic system over coordinates, momenta, or quantum states. The distribution function is the main characteristic of the most diverse (not only physical) systems that are characterized by random behavior, i.e. random change in the state of the system and, accordingly, its parameters. Even under stationary external conditions, the state of the system itself can be such that the result of measuring some of its parameters is a random variable. The distribution function in the overwhelming majority of cases contains all possible and therefore exhaustive information about the properties of such systems. In the mathematical theory of probability and mathematical statistics, the distribution function and the probability density differ from each other, but are uniquely related. Below, we will deal almost exclusively with the probability density, which (according to the long tradition accepted in physics) is called the probability distribution density or distribution function, putting an equal sign between these two terms. Random behavior is to some extent characteristic of all quantum mechanical systems: elementary particles, atoms of a molecule, etc. However, random behavior is not a specific feature of only quantum mechanical systems; many purely classical systems have this property. When throwing a coin on a hard horizontal surface, it is not clear how it will fall: with a number up or with a coat of arms. It is known that the probabilities of these events, under certain conditions, are equal to 1/2. When throwing a die, it is impossible to say with certainty which of the six numbers will be on the top face. The probability of falling out of each of the numbers under certain assumptions (a bone - a homogeneous cube without chipped edges and vertices falls on a hard, smooth horizontal surface) is 1/6. The chaotic motion of molecules is most pronounced in a gas. Even in stationary external conditions, the exact values of macroscopic parameters fluctuate (change randomly), and only their average values are constant. The description of macroscopic systems in the language of average values of macroparameters is the essence of the thermodynamic description (). Let there be an ideal monatomic gas and its three (not yet averaged) macroscopic parameters: N is the number of atoms moving inside the vessel occupied by the gas; P is the gas pressure on the vessel wall and is the internal energy of the gas. A gas is ideal and monatomic, so its internal energy is simply the sum of the kinetic energies of the translational motion of gas atoms. Number N fluctuates, at least due to the process of sorption (sticking to the wall of the vessel upon impact with it) and desorption (the process of detachment, when the molecule is detached from the wall by itself or as a result of another molecule hitting it), and finally, the process of cluster formation - short-lived complexes of several molecules. If you could measure N instantly and accurately, then the resulting dependence N(t) would be similar to the one shown in the figure. The range of fluctuations in the figure is strongly overestimated for clarity, but with a small average value (b N c ~ 10 2) the number of particles in the gas, it will be approximately the same. If we choose a small area on the vessel wall to measure the force acting on this area as a result of impacts of gas molecules in the vessel, then the ratio of the average value of the component of this force normal to the area to the area of the area is commonly called pressure. At different moments of time, a different number of molecules will fly up to the site, and at different speeds. As a result, if it were possible to measure this force instantly and accurately, there would be a picture similar to the one shown in the figure, you only need to change the designations along the vertical axis: N(t) YU P(t) and b N(t) with Yu b P(t)With. Almost all the same is true for the internal energy of the gas, only the processes leading to random changes in this amount are different. For example, flying up to the wall of a vessel, a gas molecule collides not with an abstract absolutely elastic and specularly reflecting wall, but with one of the particles that make up the material of this wall. Let the wall be steel, then these are iron ions oscillating around the equilibrium positions - the nodes of the crystal lattice. If a gas molecule flies up to the wall at that phase of the ion oscillations when it moves towards it, then as a result of the collision, the molecule will fly away from the wall with a speed greater than it flew up. Together with the energy of this molecule, the internal energy of the gas also increases. E. If a molecule collides with an ion moving in the same direction as it, then this molecule will fly off with a speed less than the one with which it flew. Finally, a molecule can get into an interstitial space (an empty space between neighboring nodes of the crystal lattice) and get stuck there, so that even strong heating cannot remove it from there. In the last two cases, the internal energy of the gas E decrease. Consequently, E(t) is also a random function of time and is the average value of this function. Having determined the position of the Brownian particle at some point in time t 1, one can accurately predict only that its position at a subsequent point in time t 2 does not exceed ( t 2 –t one)· c, where c is the speed of light in vacuum. There are cases of a discrete and continuous spectrum of states and, accordingly, a variable x. The spectrum of values of some variable is understood as the entire set of its possible values. In the case of a discrete spectrum of states, to specify the probability distribution, it is necessary, first, to indicate the full set of possible values of the random variable x 1, x 2, x 3,…x k,… (1) and second, their probabilities: W 1, W 2, W 3,…W k,… (2) The sum of the probabilities of all possible events must be equal to one (normalization condition) Description of the probability distribution by relations (1) - (3) is impossible in the case of a continuous spectrum of states and, accordingly, a continuous spectrum of possible values of the variable x. Let x takes all possible real values in the interval x O [ a, b] (4) where a and b not necessarily finite. For example, for the modulus of the velocity vector of a gas molecule VО lying within the entire range of possible values, i.e. x O [ x,x+ D x] O [ a, b] (5) Then the probability D W(x, D x) hits x in the interval (5) is equal to Here N is the total number of measurements x, and D n(x, D x) is the number of results that fall into the interval (5). Probability D W naturally depends on two arguments: x– positions of the interval inside [ a, b] and D x is its length (it is assumed, although it is not necessary at all, that D x> 0). For example, the probability of getting the exact value x, in other words, the probability of hitting x into an interval of zero length is the probability of an impossible event and therefore equals zero: D W(x, 0) = 0 On the other hand, the probability of getting the value x somewhere (doesn't matter where) within the entire interval [ a, b] is the probability of a certain event (something always happens) and therefore is equal to one (it is assumed that b > a):D W(a, b – a) = 1. Let D x few. The criterion of sufficient smallness depends on the specific properties of the system described by the probability distribution D W(x, D x). If D x small, then the function D W(x, D x) can be expanded in a series in powers of D x: If we draw a dependency graph D W(x, D x) from the second argument D x, then replacing the exact dependence with the approximate expression (7) means replacing (in a small area) the exact curve with a piece of parabola (7). In (7), the first term is exactly equal to zero, the third and subsequent terms, if D is sufficiently small, x can be omitted. Introduction of notation gives an important result D W(x, D x) » r( x) D x (8) Relation (8), which is more accurate, the smaller D x means that for a short interval, the probability of falling into this interval is proportional to its length. You can still go from a small but final D x to formally infinitesimal dx, with the simultaneous replacement of D W(x, D x) on the dW(x). Then the approximate equality (8) turns into the exact one dW(x) = r( x)· dx(9) Proportionality coefficient r( x) has a simple meaning. As can be seen from (8) and (9), r( x) is numerically equal to the probability of hitting x into an interval of unit length. Therefore, one of the names of the function r( x) is the probability distribution density for the variable x. Function r( x) contains all the information about how the probability dW(x) hits x in the interval of a given length dx depends on the location of this interval, i.e. it shows how the probability is distributed over x. Therefore, the function r( x) is commonly called the distribution function for the variable x and, thus, the distribution function for that physical system, for the sake of describing the spectrum of states of which the variable was introduced x. The terms "probability density" and "distribution function" are used interchangeably in statistical physics. We can consider a generalization of the definition of probability (6) and distribution function (9) to the case, for example, of three variables. Generalization to the case of an arbitrarily large number of variables is carried out in exactly the same way. Let the state of a physical system randomly varying in time be determined by the values of three variables x, y and z with continuous spectrum: x O [ a, b] y O [ c, d] z O [ e, f] (10) where a, b,…, f, as before, are not necessarily finite. Variables x, y and z can be, for example, the coordinates of the center of mass of a gas molecule, the components of its velocity vector x YU Vx, y YU V y and z YU Vz or impulse, etc. An event is understood as the simultaneous occurrence of all three variables in intervals of length D x, D y and D z respectively, i.e.: x O [ x, x+ D x] y O [ y, y+ D y] z O [ z, z+ D z] (11) The probability of an event (11) can be determined similarly to (6) with the difference that now D n– number of measurements x, y and z, whose results simultaneously satisfy relations (11). Using a series expansion similar to (7) gives dW(x, y, z) = r( x, y, z)· dx dy dz(13) where r( x, y, z) is the distribution function for three variables at once x, y and z. In the mathematical theory of probability, the term "distribution function" is used to denote a quantity different from r( x), namely: let x be some value of a random variable x. The function Ф(x), which gives the probability that x takes a value no greater than x and is called the distribution function. The functions r and Ф have different meanings, but they are related. Using the probability addition theorem gives (here a is the left end of the range of possible values x (cm. PROBABILITY THEORY: , (14) whence Using the approximate relation (8) gives D W(x, D x) » r( x) D x. Comparison with the exact expression (15) shows that using (8) is equivalent to replacing the integral in (16) with the product of the integrand r( x) by the length of the integration interval D x: Relation (17) will be exact if r = const, therefore, the error when replacing (16) with (17) will be small when the integrand changes slightly over the length of the integration interval D x. You can enter D x eff is the length of the interval on which the distribution function r( x) changes significantly, i.e. by a value of the order of the function itself, or the quantity Dr eff modulo order r. Using the Lagrange formula, we can write: whence it follows that D x eff for any function r The distribution function can be considered "almost constant" over a certain interval of change of the argument if its increment |Dr| on this interval, the absolute value is much less than the function itself at the points of this interval. Requirement |Dr| eff| ~ r (distribution function r і 0) gives D x x eff (20) the length of the integration interval should be small compared to the one on which the integrand changes significantly. The illustration is fig. one. The integral on the left side of (17) is equal to the area under the curve. The product on the right side of (17) is the area of the shaded in Fig. 1 column. The criterion for the smallness of the difference between the corresponding areas is the fulfillment of inequality (20). This can be verified by substituting into the integral (17) the first terms of the expansion of the function r( x) in a series in powers The requirement that the correction (the second term on the right-hand side of (21) be compared with the first one be small gives inequality (20) with D x eff from (19). Examples of a number of distribution functions that play an important role in statistical physics. Maxwell distribution for the projection of the velocity vector of a molecule onto a given direction (for example, this is the direction of the axis OX). Here m is the mass of a gas molecule, T- its temperature k is the Boltzmann constant. Maxwell distribution for the modulus of the velocity vector: Maxwell distribution for the energy of translational motion of molecules e = mV 2/2 Boltzmann distribution, more precisely, the so-called barometric formula, which determines the distribution of the concentration of molecules or air pressure in height h from some “zero level” under the assumption that the air temperature does not depend on height (isothermal atmosphere model). In fact, the temperature in the lower layers of the atmosphere drops noticeably with increasing altitude.Changing Variables Technique
Generalization for the reduce function
Distribution functions
Random variables and distribution functions
Bulk Functions
Independent random variables
Simulation of random variables
Illustrating Probability Transformation
Exponential function and its variables
General population and random variable
distribution function
Probability density calculation using MS EXCEL functions
Calculation of probabilities using MS EXCEL functions
Instead of +∞, the value 9.999E+307= 9.999*10^307 is entered into the formula, which is the maximum number that can be entered in a MS EXCEL cell (so to speak, the closest to +∞).
Examples.
Brownian motion.
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