A non-negative random variable has gamma distribution, if its distribution density is expressed by the formula

where and , is the gamma function:

Thus, gamma distribution is a two-parameter distribution, it occupies an important place in mathematical statistics and reliability theory. This distribution is limited on the one hand.

If the distribution curve shape parameter is an integer, then the gamma distribution describes the time required for the occurrence of events (failures), provided that they are independent and appear with a constant intensity .

In most cases, this distribution describes the operating time of the system with redundant failures of aging elements, the recovery time of the system with redundant failures of aging elements, the recovery time of the system, etc. For various quantitative values ​​of the parameters, the gamma distribution takes on a wide variety of forms, which explains its wide application. .

The probability density of the gamma distribution is defined by the equality if

distribution function. (9)

Note that the reliability function is expressed by the formula:

The gamma function has the following properties: , , (11)

whence it follows that if is a non-negative integer, then

In addition, in the future we will need one more property of the gamma function: ; . (13)

Example. Recovery of electronic equipment obeys the law of gamma distribution with parameters and . Determine the probability of restoring the hardware per hour.

Solution. To determine the recovery probability, we use the formula (9) .

For integer positive values functions , and for .

If we pass to new variables, the values ​​of which will be expressed ; , then we get the table integral:

In this expression, the solution of the integral on the right side can be determined by the same formula:

and when will

For and , the new variables will be equal to and , and the integral itself will be equal to

The value of the function will be

Let's find the numerical characteristics of a random variable , subject to the gamma distribution

In accordance with equality (13), we obtain . (14)

The second initial moment is found by the formula

where . (15)

Note that at , the failure rate decreases monotonically, which corresponds to the run-in period of the product. With , the failure rate increases, which characterizes the period of wear and aging of the elements.

At , the gamma distribution coincides with the exponential distribution; at , the gamma distribution approaches the normal law. If it takes the values ​​of arbitrary positive integers, then such a gamma distribution is called Erlang distribution of the th order:

Here it suffices to point out that the Erlang law -th order the sum of independent random variables, each of which is distributed according to an exponential law with a parameter . Erlang's law th order is closely related to the stationary Poisson (simplest) flow with intensity .

Indeed, let there be such a flow of events in time (Fig. 6).

Rice. 6. Graphical representation of the Poisson flow of events in time

Consider a time interval consisting of the sum intervals between events in such a stream. It can be proved that a random variable will obey the Erlang law -th order.

Distribution density of a random variable distributed according to the Erlang law th order, can be expressed in terms of a tabular Poisson distribution function:

If the value multiple of and , then the gamma distribution coincides with the chi-square distribution.

Note that the distribution function of a random variable can be calculated using the following formula:

where are determined by expressions (12) and (13).

Therefore, we have the following equalities, which will be useful to us in what follows:

Example. The flow of products produced on the conveyor is the simplest with the parameter . All manufactured products are controlled, defective ones are placed in a special box, which contains no more than products, the probability of marriage is . Determine the distribution law for the time of filling the box with defective products and the value , based on the fact that the box is not likely to overflow during the shift.

Solution. The intensity of the simplest flow of defective products will be . It is obvious that the time of filling the box with defective products is distributed according to the Erlang law

with parameters and :

hence (18) and (19): ; .

The number of defective items over time will be distributed according to Poisson's law with the parameter . Therefore, the desired number must be found from the condition. (20)

For example, at [product/h]; ; [h]

from the equation for

A random variable with an Erlang distribution has the following numerical characteristics(Table 6).

Table 6

Note that a random variable with a normalized Erlang distribution of the th order has the following numerical characteristics (Table 7).

Table 7

This article describes the formula syntax and usage of the function GAMMA.DIST in Microsoft Excel.

Returns the gamma distribution. This function can be used to examine variables that have a skewed distribution. The gamma distribution is widely used in the analysis of queuing systems.

Syntax

GAMMA.DIST(x,alpha,beta,cumulative)

The GAMMA.DIST function syntax has the following arguments:

x is a required argument. The value for which you want to calculate the distribution.

Alpha is a required argument. Distribution parameter.

Beta is a required argument. Distribution parameter. If beta = 1, GAMMA.DIST returns the standard gamma distribution.

Integral is a required argument. Boolean value that defines the form of the function. If the cumulative argument is TRUE, GAMMA.DIST returns the cumulative distribution function; if this argument is FALSE, the probability density function is returned.

Remarks

Example

Copy the sample data from the following table and paste it into cell A1 of a new Excel sheet. To display formula results, select them and press F2 followed by ENTER. Change the width of the columns, if necessary, to see all the data.

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

Examples evenly distributed quantities are rounding errors. 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 , physical meaning which is the average number of failures per unit of time, excluding system downtime for repairs.

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 normal distribution 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. So, errors in the execution of commands by an automated device, output errors spaceship 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, the random variables formed by the summation a large number 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.

Consider the Gamma distribution, calculate it expected value, dispersion, mode. Using the MS EXCEL GAMMA.DIST() function, we plot the distribution function and probability density graphs. Let's generate an array of random numbers and estimate the distribution parameters.

Gamma distribution(English) Gammadistribution) depends on 2 parameters: r(defines the shape of the distribution) and λ (defines the scale). this distribution is given by the following formula:

where Г(r) is the gamma function:

if r is a positive integer, then Г(r)=(r-1)!

The above entry form distribution density clearly shows its relationship with. For r=1 Gamma distribution boils down to exponential distribution with parameter λ.

If the parameter λ is an integer, then Gamma distribution is the sum r independent and equally distributed exponential law with parameter λ of random variables x. So the random variable y= x 1 + x 2 +… x r It has gamma distribution with parameters r and λ.

, in turn, is closely related to discrete . If Poisson distribution describes the number of random events that occurred in a certain time interval, then exponential distribution, in this case, describes the length of the time interval between two successive events.

It follows from this that, for example, if the time before the first event is described by exponential distribution with the parameter λ, then the time until the second event occurs is described gamma distribution with r = 2 and the same parameter λ.

Gamma distribution in MS EXCEL

In MS EXCEL, an equivalent, but different, form of notation is adopted density gamma distribution.

Parameter α ( alpha) is equivalent to the parameter r, and the parameter b (beta) - parameter 1/λ. Below we will adhere to just such a notation, since this will make it easier to write formulas.

In MS EXCEL, starting from version 2010, for Distribution Gamma there is a GAMMA.DIST() function, English name- GAMMA.DIST(), which allows you to calculate probability density(see formula above) and (probability that a random variable X having gamma distribution, takes a value less than or equal to x).

Note: Prior to MS EXCEL 2010, EXCEL had a GAMMADIST() function that allows you to calculate integral distribution function And probability density. GAMMADIST() is left in MS EXCEL 2010 for compatibility.

Function Graphs

The example file contains graphs probability distribution density And integral distribution function.

Gamma distribution has the designation Gamma (alpha; beta).

Note: For the convenience of writing formulas in the example file for distribution parameters alpha and beta created corresponding .

Note: Dependence on 2 parameters allows you to build distributions of various shapes, which expands the application of this distribution. Gamma distribution, like Exponential Distribution often used to calculate the waiting time between random events. In addition, it is possible to use this distribution for precipitation modeling and road design.

As shown above, if the parameter alpha= 1, then the GAMMA.DIST() function returns with the parameter 1/beta. If the parameter beta= 1, the GAMMA.DIST() function returns the standard gamma distribution.

Note: Because is a special case gamma distribution, then the formula =GAMMA.DIST(x,n/2,2,TRUE) for a positive integer n returns the same result as the formula =XI2.DIST(x, n, TRUE) or =1-XI2.DIST.X(x;n) . And the formula =GAMMA.DIST(x,n/2,2,FALSE) returns the same result as the formula =XI2.DIST(x, n, FALSE), i.e. probability density XI2 distributions.

IN example file on the Graphics sheet calculation is given gamma distribution equal alpha*beta And

BASIC LAWS OF DISTRIBUTION OF CONTINUOUS RANDOM VARIABLES

Hthe normal distribution law and its significance in probability theory. Logarithmically normal law. Gamma distribution. Exponential law and its use in reliability theory, queuing theory. Equal law. distribution. Student distribution. Fisher distribution.

1. Normal distribution law (Gauss law).

The probability density of a normally distributed random variable is expressed by the formula:

. (8.1)

On fig. 16 shows the distribution curve. It is symmetrical about

Rice. 16 Fig. 17

points (maximum point). When decreasing, the ordinate of the maximum point increases indefinitely. In this case, the curve is proportionally flattened along the x-axis, so that its area under the graph remains equal to one(Fig. 17).

The normal distribution law is very widespread in practical problems. Lyapunov was the first to explain the reasons for the wide spread of the normal distribution law. He showed that if a random variable can be considered as the sum of a large number of small terms, then under sufficiently general conditions the law of distribution of this random variable is close to normal, regardless of what the laws of distribution of individual terms are. And since practically random variables in most cases are the result of a large number of different causes, the normal law turns out to be the most common distribution law (for more on this, see Chapter 9). Let us indicate the numerical characteristics of a normally distributed random variable:

Thus, the parameters and in expression (8.1) of the normal distribution law represent the mathematical expectation and the standard deviation of a random variable. Taking this into account, formula (8.1) can be rewritten as follows:

.

This formula shows that the normal distribution law is completely determined by the mathematical expectation and the variance of a random variable. Thus, the mathematical expectation and variance completely characterize a normally distributed random variable. It goes without saying that in the general case, when the nature of the distribution law is unknown, knowledge of the mathematical expectation and variance is not enough to determine this distribution law.

Example 1. Calculate the probability that a normally distributed random variable satisfies the inequality .

Solution. Using property 3 of the probability density (Chapter 4, Section 4), we get:

.

,

where is the Laplace function (see Appendix 2).

Let's do some numerical calculations. If we put , under the conditions of Example 1, then

The last result means that with a probability close to one (), a random variable obeying the normal distribution law does not go beyond the interval . This statement is called three sigma rules.

Finally, if , , then a random variable distributed according to the normal law with such parameters is called a standardized normal variable. On fig. 18 shows a graph of the probability density of this value .

2. Logarithmically normal distribution.

A random variable is said to have a log-normal distribution (abbreviated lognormal distribution) if its logarithm is normally distributed, i.e., if

where the value has a normal distribution with parameters , .

The density of the lognormal distribution is given by the following formula:

, .

The mathematical expectation and variance are determined by the formulas

,

.

The distribution curve is shown in fig. 19.

The log-normal distribution occurs in a number of technical problems. It gives the distribution of particle sizes during crushing, the distribution of the contents of elements and minerals in igneous rocks, the distribution of the abundance of fish in the sea, etc. It is found in all

those problems where the logarithm of the quantity under consideration can be represented as the sum of a large number of independent uniformly small quantities:

,

i.e. , where are independent.