Probability the opposite events(failure-free operation over time t) is equal to R(t) = P(T>t) = 1 - F(t).

Definition. Reliability functionR(t) is a function that determines the probability of failure-free operation of a device over time t.

Often on practice The duration of failure-free operation is subject to an exponential distribution law.

At all speaking, If consider a new device, then the probability of failure at the beginning of its operation will be greater, then the number of failures will decrease and will have almost the same value for some time. Then (when the device exhausts its resource) the number of failures will increase.

Others words, we can say that the functioning of a device throughout its entire existence (in terms of the number of failures) can be described by a combination of two exponential laws (at the beginning and end of operation) and a uniform distribution law.

The reliability function for any device under the exponential distribution law is equal to:

This ratio is called exponential law of reliability.

Important property, which makes it possible to significantly simplify the solution of reliability theory problems, is that the probability of failure-free operation of a device over a time interval t does not depend on the time of previous work before the start of the interval under consideration, but depends only on the duration of time t.

So way, the failure-free operation of the device depends only on the failure rate l and does not depend on the failure-free operation of the device in the past.

Since it has a similar property only an exponential distribution law, then this fact allows us to determine whether the distribution law of a random variable is exponential or not.

2.8 Chi-square distribution

Let X i (i=1,2,…,n)- normal independent random variables, and the mathematical expectation of each of them is equal to zero, and the standard deviation is equal to one. Then the sum of the squares of these quantities

distributed according to the law (“Chi-square”) with k=n degrees of freedom; if these quantities are related by one linear relationship, for example, then the number of degrees of freedom k=n-1.

The density of this distribution

Where -Gamma function; in particular,

From here it is seen that the Chi-square distribution is determined by one parameter - the number of degrees of freedom k. As the number of degrees of freedom increases, the distribution slowly approaches normal.

2.9 Student distribution

Let Z be a normal random variable, with M(Z)=0, s(Z)=1, and V be a variable independent of Z, which is distributed according to the law with k degrees of freedom. Then the value

has a distribution called the t-distribution or Student distribution, k degrees of freedom. So the ratio is normalized normal size to the square root of an independent random variable distributed according to the law

« Chi-square with k degrees of freedom, divided by k, divided by k is distributed according to Student's law with k degrees of freedom. . As the number of degrees of freedom increases, the distribution slowly approaches normal.

2.9 Normal distribution law

Definition. Normal is the probability distribution of a continuous random variable, which is described by the probability density

The normal distribution law is also called Gauss's law.

The normal distribution law occupies a central place in probability theory. This is due to the fact that this law manifests itself in all cases when a random variable is the result of the action of a large number various factors. All other distribution laws approach the normal law.

Can easily show that the parameters and included in the distribution density are, respectively, the mathematical expectation and the standard deviation of the random variable X.

Let's find the distribution function F(x).

The density plot of a normal distribution is called a normal curve or Gaussian curve.

A normal curve has the following properties:

1 ) The function is defined on the entire number line.

2 ) In front of everyone X the distribution function takes only positive values.

3 ) The OX axis is the horizontal asymptote of the probability density graph, because with unlimited increase in the absolute value of the argument X, the value of the function tends to zero.

4 ) Let's find the extremum of the function.

Because at y’ > 0 at x< m And y'< 0 at x > m, then at the point x = t the function has a maximum equal to .

5 ) The function is symmetrical with respect to the straight line x = a, because difference

(x - a) is included in the squared distribution density function.

6 ) To find the inflection points of the graph, we find the second derivative of the density function.

At x = m+s and x = m- s the second derivative is equal to zero, and when passing through these points it changes sign, i.e. at these points the function has an inflection point.

A continuous random variable has exponential (exponential )distribution law with parameter if its probability density has the form:

(12.1)

This is a constant positive value. That. exponential distribution is determined by one positive parameter . Let's find the integral function of the exponential distribution:

(12.3)

Rice. 12.1. Differential exponential distribution function ()

Rice. 12.2. Cumulative exponential distribution function ()

Numerical characteristics of exponential distribution

Let's calculate the mathematical expectation and variance of the exponential distribution:

To calculate the variance, we will use one of its properties:

Because , then it remains to calculate:

Substituting (12.6) into (12.5), we finally obtain:

(12.7)

For a random variable distributed according to the exponential law, the mathematical expectation is equal to the standard deviation.

Example 1. Write the differential and integral functions of the exponential distribution if the parameter .

Solution. a) The distribution density has the form:

b) The corresponding integral function is equal to:

Example 2. Find the probability of falling into a given interval for a SV distributed according to an exponential law

Solution. Let's find a solution, remembering that: . Now, taking into account (12.3), we obtain:

Reliability function

We will call some device an element, regardless of whether it is “simple” or “complex”. Let the element begin to work at the moment of time, and after a period of time a failure occurs. Let us denote by continuous SV the duration of the element’s failure-free operation. If the element operates without failure (before a failure occurs) for a time less than , then, consequently, a failure will occur during the duration. Thus, probability of failure over time duration is determined by the integral function:

. (12.8)

Then the probability of failure-free operation for the same time duration is equal to the probability of the opposite event, i.e.

Reliability functionis a function that determines the probability of failure-free operation of an element over a period of time.

Often the duration of failure-free operation of an element has an exponential distribution, the integral function of which is equal to:

. (12.10)

Then, in the case of exponential distribution of the element’s failure-free operation time and taking into account (12.9), the reliability function will be equal to:

. (12.11)

Example 3. The failure-free operation time of the element is distributed according to the exponential law at (time in hours). Find the probability that the element will operate without failure for 100 hours.

Solution. In our example, then we will use (12.11):

The exponential reliability law is very simple and convenient for solving practical problems. This law has the following important property:

The probability of failure-free operation of an element over a time interval of length does not depend on the time of previous operation before the start of the interval under consideration, but depends only on the duration of time(at a given failure rate).

Let us prove this property by introducing the following notation:

failure-free operation of the element over an interval of ;

Then the event is that the element operates without failure for an interval of duration . Let us find the probabilities of these events using formula (12.11), assuming that the failure-free operation time of the element is subject to the exponential law:

Let's find the conditional probability that an element will work without failure in a time interval, provided that it has already worked without failure in the previous time interval:

(12.13)

We see that the resulting formula does not depend on , but only on . Comparing (12.12) and (12.13), we can conclude that the conditional probability of failure-free operation of an element in an interval of duration , calculated under the assumption that the element worked without failure in the previous interval, is equal to the unconditional probability.

So, in the case of the exponential reliability law, the failure-free operation of an element “in the past” does not affect the probability of its failure-free operation “in the near future.”

Elements of combinatorics

Space of elementary events. Random events.

Probability

Modern concept of probability

Classical probabilistic scheme

Geometric probabilities

Law of addition of probabilities

Probability multiplication theorem

Total Probability Formula

Theorem of hypotheses. Bayes' formula.

Repetition of tests. Bernoulli scheme.

Local theorem of Moivre-Laplace

Moivre-Laplace integral theorem

Poisson's Theorem (Law of Rare Events)

Random variables

Distribution functions

Continuous random variable and distribution density

Basic properties of distribution density

Numerical characteristics of a one-dimensional random variable

Properties of mathematical expectation

Moments of a random variable

Dispersion properties

Skewness and kurtosis

Multivariate random variables

Properties of the two-dimensional distribution function

Probability density of a two-dimensional random variable

Buffon's problem

Conditional distribution density

Numerical characteristics of a system of random variables

Properties of the correlation coefficient

Normal (Gaussian) distribution law

Probability of hitting the interval

Properties of the normal distribution function

Distribution (chi-square)

Exponential distribution law

Numerical characteristics of exponential distribution

Reliability function

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.

The mathematical 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

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:

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 useful life described by 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.