Example file

Let's consider the Exponential distribution, calculate its mathematical expectation, variance, and median. Using the MS EXCEL function EXP.DIST(), we will construct graphs of the distribution function and probability density. Let's generate an array of random numbers and estimate the distribution parameter.

(English) Exponentialdistribution) often used to calculate the waiting time between random events. Below are situations where it can be used Exponential distribution :

• Time intervals between the appearance of visitors in the cafe;
• The time intervals of normal operation of equipment between the occurrence of malfunctions (faults arise due to random external influences, and not due to wear, see);
• Time spent serving one customer.

Random number generation

To generate an array of numbers distributed over exponential law, you can use the formula =-LN(RAND())/ λ

The RAND() function generates from 0 to 1, which exactly corresponds to the range of probability changes (see. example file sheet Generation).

If the random numbers are in the range B14:B213 , then the parameter estimate exponential distribution λ can be done using the formula =1/AVERAGE(B14:B213) .

Tasks

Exponential distribution widely used in the discipline of Reliability Engineering. Parameter λ called failure rate, A 1/ λ – average time to failure .

Suppose that an electronic component of a certain system has a lifespan beneficial use, described Exponential distribution With failure rate equal to 10^(-3) per hour, thus λ = 10^(-3). Average time to failure equals 1000 hours. To calculate the probability that a component will fail in Average time to failure then you need to write the formula:

Those. the result does not depend on the parameter λ .

In MS EXCEL the solution looks like this: =EXP.DIST(10^3, 10^(-3), TRUE)

Task . Average time to failure some component is equal to 40 hours. Find the probability that the component will fail between 20 and 30 hours of operation. =EXP.DIST(30, 1/40, TRUE)- EXP.DIST(20, 1/40, TRUE)

ADVICE: You can read about other MS EXCEL distributions in the article.

Exponential (exponential) distribution

Let's consider a family of distributions widely used in management decision-making and other applied research - the family of exponential distributions. Let's analyze the probability!! model leading to such distributions. To do this, consider the “flow of events”, i.e. a sequence of events occurring one after another at certain points in time. Examples include: the uptime of a computer system, the interval between successive arrivals of cars at the stop line of an intersection, the flow of customer requests to a bank branch; the flow of buyers applying for goods and services; call flow at the telephone exchange; flow of equipment failures in the technological chain, etc.

In the theory of event flows, the summation theorem of event flows is valid. The total flow consists of a large number of independent partial flows, none of which has a dominant influence on the total flow. Thus, the flow of calls entering a telephone exchange consists of a large number of independent call flows emanating from individual subscribers. In the case when the characteristics of the flows do not depend on time, the total flow is completely described by one number X- flow intensity. For the total flow, the distribution function of the random variable X- the length of the time interval between successive events has the following form:

This distribution is called exponential (exponential) distribution. This function sometimes includes a shift parameter c.

The exponential distribution has only one parameter, which determines its characteristics. The distribution density has the following form:

Where X- constant positive value.

Graph of a function /(X) shown in Fig. 9.12.

Rice. 9.12.

In Fig. Figure 9.13 shows a graph of the density of the exponential distribution for different parameters X.

The exponential distribution characterizes the distribution of time between independent events that occur with constant intensity. The exponential law is characteristic of the distribution of random variables, the change of which is due to the influence of some dominant factor. In reliability theory, this distribution describes the distribution of sudden failures, since the latter are rare events. The exponential distribution also serves to describe

Rice. 9.13. Density of exponential distribution for different parameters X

operating time of complex systems that have undergone a run-in period, and to describe the uptime of a system with a large number of series-connected elements, each of which does not have a large impact on system failure.

Theoretical frequencies for the exponential distribution law are determined by the formula

Where N- volume of the population; 1g to- interval length; e- the base of the natural logarithm; X- conditional deviations of class averages:

Let's consider the alignment of the empirical distribution (Table 9.4) according to the exponential law.

Table 9.4

Empirical frequencies to smooth out the exponential distribution

We have N= 160; b k = 41; x = 54.59. Calculation of values ​​of conditional deviations of class means, auxiliary values e _1 and theoretical frequencies are produced in table. 9.5.

Table 95

Alignment of empirical frequencies according to the exponential law

Empirical data X	Empirical frequency T			Theoretical frequencies

We depict the empirical and theoretical frequencies of the exponential distribution graphically in Fig. 9.14.

The exponential distribution is a special case of the Weibull-Gnedenko distribution (corresponding to the value of the form parameter b = 1).

Where λ – constant positive value.

From expression (3.1), it follows that the exponential distribution is determined by one parameter λ.

This feature of the exponential distribution points to him advantage over distributions , depending on a larger number of parameters. Usually the parameters are unknown and we have to find their estimates (approximate values), of course, It’s easier to evaluate one parameter than two or three, etc. . An example of a continuous random variable distributed according to the exponential law , can serve as the time between the occurrences of two consecutive events of the simplest flow.

Let's find the distribution function of the exponential law .

So

Density graphs and distribution functions of the exponential law are shown in Fig. 3.1.

Considering that we get:

The function values ​​can be found from the table.

Numerical characteristics of exponential distribution

Let a continuous random variableΧ distributed according to exponential law

Let's find the mathematical expectation , using the formula for calculating it for a continuous random variable:

Hence:

Let's find the standard deviation , for which we extract the square root of the variance:

Comparing (3.4), (3.5) and (3.6), it is clear that

i.e.the mathematical expectation and standard deviation of the exponential distribution are equal.

The exponential distribution is widely used in various applications of financial and technical problems, for example, in reliability theory.

4. Chi-square and Student distributions.

4.1 Chi-square distribution (- distribution)

Let Χ i (ί = 1, 2, ..., n) be normal independent random variables , and the mathematical expectation of each of them is zero , A standard deviation - unit .

Then the sum of the squares of these quantities

distributed according to lawWithdegrees of freedom , if these quantities are related by one linear relationship, for example, then the number of degrees of freedom

The chi-square distribution has found widespread use in mathematical statistics.

The density of this distribution

where is the gamma function, in particular .

This shows that the chi-square distribution is determined by one parameter - number of degrees of freedomk.

As the number of degrees of freedom increases, the chi-square distribution slowly approaches normal.

The chi-square distribution is obtained if Erlang's distribution law is taken to be λ = ½ And k = n /2 – 1.

Expectation and variance of a random variable with a chi-square distribution, are determined by simple formulas, which we present without derivation:

From the formula it follows that atThe chi-square distribution coincides with the exponential distribution whenλ = ½ .

The cumulative distribution function for the chi-square distribution is determined through special incomplete tabulated gamma functions

In Fig. 4.1. given graphs of the probability density and distribution function of a random variable having a chi-square distribution for n = 4, 6, 10.

Fig.4.1. A )Probability density graphs with chi-square distribution

Fig.4.1. b) Graphs of the distribution function with chi-square distribution

4.2 Student distribution

Let Z be a normal random variable, and

A V – a value independent of Z, which is distributed according to the chi-square law withk degrees of freedom. Then size:

has a distribution calledt -distribution or Student distribution (pseudonym of the English statistician W. Gosset),

Withk = n- 1 degrees of freedom (n - the volume of statistical sampling when solving statistical problems).

So , the ratio of the normalized normal value to the square root of an independent random variable distributed according to the chi-square law with k degrees of freedom , divided by k, distributed according to Student's law with k degrees of freedom.

Student distribution density:

The random variable has uniform distribution, if the probability that it takes any value in the interval limited by the minimum number A and the maximum number b, constant. Since the density plot of this distribution has the shape of a rectangle, the uniform distribution is sometimes called rectangular (see panel B in Fig. 1).

Rice. 1. Three continuous distributions

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The uniform distribution density function is given by the formula:

Where A- minimum value of the variable X, b- maximum value of the variable X.

Mathematical expectation of uniform distribution:

(2) μ = (a +b) / 2

Uniform distribution variance:

(3) σ 2 = (b – a) 2 / 12

Standard deviation of uniform distribution:

The most common use of uniform distribution is to select random numbers. When performing a simple random selection, it is assumed that each number is drawn from a population uniformly distributed in the interval from 0 to 1. Let's calculate the probability of drawing a random number greater than 0.1 and less than 0.3.

The graph of the uniform distribution density function for a = 0 and b = 1 is shown in Fig. 2. The total area of ​​the rectangle bounded by this function is equal to one. Consequently, this graph satisfies the requirement that the area of ​​the figure bounded by the density graph of any distribution must be equal to one. The area of ​​a rectangle enclosed between the numbers 0.1 and 0.3 is equal to the product of the lengths of its sides, i.e. 0.2 x 1 = 0.2. So, P(0.1< X < 0,3) = 0,2 х 1 = 0,2.

Rice. 2. Density graph of uniform distribution; calculating the probability P(0.1< X < 0,3) для равномерного распределения при а = 0 и b = 1

The expected value, variance and standard deviation of a uniform distribution at a = 0 and b = 1 are calculated as follows:

Let's look at an example. Let us assume that the moments of failure of the device for monitoring air purity are evenly distributed throughout the day.

1. On a certain day, daylight begins at 5:55 and ends at 19:38. What is the probability that a device hardware failure will occur during daylight hours?
2. Let's say that from 22:00 to 5:00 the device goes into low power mode. What is the probability that failure will occur within the specified time period?
3. Let's assume that the device includes a processor that checks the functionality of the equipment every hour. What is the probability that a failure will be detected within 10 minutes?
4. Let's assume that the device includes a processor that checks the functionality of the equipment every hour. What is the probability that the failure will be detected no sooner than 40 minutes later?

Solution. 1. Since the problem statement states that the moments of device failure are evenly distributed throughout the day, the probability of failure during daylight hours is a fraction of this time of day. P (failure during daylight hours) = 19:38 – 5:55 = 57.2%. For calculations, see attached Excel file. If we imagine the difference between the end and beginning of daylight hours in percentage format, we get the answer - 57.2%. The trick is that in Excel, a day is a unit, one hour is 1/24, so a time interval less than a day will be a percentage of that day.

2. P (failure from 22:00 to 5:00) = 2:99 + 5:00 = 29.2%.

3. P (failure detection no later than 10 minutes) = 10 / 60 = 16.7%

4. P (failure detection no earlier than 40 minutes) = (60 – 40) / 60 = 33.3%

Exponential distribution

The exponential distribution is continuous, positively skewed, and ranges from zero to plus infinity (see panel B of Figure 1). The exponential distribution proves to be quite useful in business applications, especially in modeling manufacturing and queuing systems. It is widely used in scheduling theory to model the time intervals between two requests, which could be a customer arriving at a bank or fast food restaurant, a patient entering a hospital, or visiting a Web site.

The exponential distribution depends on only one parameter, which is denoted by the letter λ and represents the average number of requests entering the system per unit of time. Magnitude 1/λ equal to the average time elapsed between two consecutive requests. For example, if the system receives an average of 4 requests per minute, i.e. λ = 4, then the average time elapsed between two consecutive requests is 1/λ= 0.25 min, or 15 s. The probability that the next request will arrive sooner than in X units of time is determined by formula (5).

(5) P (request arrival time< X) = 1 – e –λ x

Where e- natural logarithm base equal to 2.71828, λ – the average number of requests entering the system per unit of time, X– value of a continuous quantity, 0< X < ∞.

Let's illustrate the use of exponential distribution with example 2. Let's assume that 20 clients come to a bank branch per hour. Let's assume that one client has already arrived at the bank. What is the probability that the next customer will arrive within 6 minutes? IN in this caseλ = 20, X = 0.1 (6 min = 0.1 h). Using formula (5), we obtain:

P(arrival time of the second client< 0,1) = 1 – е –20*0,1 = 0,8647

Thus, the probability that the next client will arrive within 6 minutes is 86.47%. The exponential distribution can be calculated using the Excel function =EXP.DIST() (Figure 3).

Rice. 3. Calculation of exponential distribution using the function =EXP.DIST()

Materials from the book Levin et al. Statistics for Managers are used. – M.: Williams, 2004. – p. 379–383

Exponential distribution law also called the basic law of reliability, is often used to predict reliability during normal operation of products, when gradual failures have not yet emerged and reliability is characterized sudden failures. These failures are caused by an unfavorable combination of many circumstances and therefore have a permanent intensity. The exponential distribution is quite widely used in the theory of queuing; it describes the distribution of time between failures of complex products and the failure-free operation time of electronic equipment elements.

Let us give examples of an unfavorable combination of operating conditions for machine parts that cause their sudden failure. For a gear train, this may be the effect of the maximum load on the weakest tooth as it engages; for elements of electronic equipment - exceeding the permissible current or temperature conditions.

The distribution density of the exponential law (Fig. 1) is described by the relation

f(x) = λ e −λ x; (3)

the distribution function of this law is the relation

F(x) = 1− e −λ x; (4)

reliability function

P(x) = 1− F(x) = e −λ x; (5)

mathematical expectation of a random variable X

random variable variance X

(7)

The exponential law in reliability theory has found wide application because it is simple for practical use. Almost all problems solved in reliability theory turn out to be much simpler when using the exponential law than when using other distribution laws. The main reason for this simplification is that with an exponential law, the probability of failure-free operation depends only on the duration of the interval and does not depend on the time of previous operation.

Fig. 1. Density graph of exponential distribution

Example 2. Based on the operating data of the generator, it has been established that the time between failures obeys an exponential law with the parameter λ=2*10 -5 h -1 . Find the probability of failure-free operation over time t=100 hours. Determine the mathematical expectation of time between failures.

Solution. To determine the probability of failure-free operation, we use formula (5), according to which

The mathematical expectation of time between failures is