EntranceDistributions of continuous random variables. Normal distribution law of a continuous random variable Find an interval symmetric with respect to the mathematical deviation
The mathematical expectation a=3 and the standard deviation =5 of a normally distributed random variable X are given.
Write down the probability distribution density and schematically plot it.
Find the probability that x will take a value from the interval (2;10).
Find the probability that x will take a value greater than 10.
Find an interval symmetrical with respect to the mathematical expectation, in which the values of the quantity x will be contained with probability =0.95.
1). Let's compose the distribution density function of a random variable X with parameters а=3, =5 using the formula
. Let's construct a schematic graph of the function 
. Let us pay attention to the fact that the normal curve is symmetrical with respect to the straight line x = 3 and has max at this point equal to 
, i.e. 
and two inflection points 
with ordinate 
Let's build a graph 
2) Let's use the formula: 
The function values are found from the application table.
4) Let's use the formula 
. According to the condition, the probability of falling into an interval symmetrical with respect to the mathematical expectation 
. Using the table, we find t at which Ф(t)=0.475, t=2. Means 
. Thus, 
. The answer is x(-1;7).
To problems 31-40.
Find a confidence interval for an estimate with a reliability of 0.95 of the unknown mathematical expectation a of a normally distributed characteristic X of the general population, if the general standard deviation =5, the sample mean 
and sample size n=25.
We need to find a confidence interval 
.
All quantities except t are known. Let's find t from the ratio Ф(t)=0.95/2=0.475. Using the appendix table we find t=1.96. Substituting, we finally get the desired confidence interval of 12.04 The technical control department checked 200 batches of identical products and received the following empirical distribution, frequency n i - the number of batches containing x i non-standard products. It is required, at a significance level of 0.05, to test the hypothesis that the number of non-standard products X is distributed according to Poisson's law. Let's find the sample mean: Let us take the sample mean =0.6 as an estimate of the parameter of the Poisson distribution. Therefore, the assumed Poisson's law Setting i=0,1,2,3,4, we find the probabilities P i of the appearance of i non-standard products in 200 batches: Let's find the theoretical frequencies using the formula Let's compare the empirical and theoretical frequencies using the Pearson test. To do this, we will create a calculation table. Let's combine the small frequencies (4+2=6) and the corresponding theoretical frequencies (3.96+0.6=4.56). Without exaggeration, it can be called a philosophical law. Observing various objects and processes in the world around us, we often come across the fact that something is not enough, and that there is a norm: What examples can you give? There are simply darkness of them. This is, for example, the height, weight of people (and not only), their physical strength, mental abilities, etc. There is a "main mass" (for one reason or another) and there are deviations in both directions. These are different characteristics of inanimate objects (same size, weight). This is a random duration of processes, for example, the time of a hundred-meter race or the transformation of resin into amber. From physics, I remembered air molecules: some of them are slow, some are fast, but most move at “standard” speeds. Next, we deviate from the center by one more standard deviation and calculate the height: Marking points on the drawing (green) and we see that this is quite enough. At the final stage, carefully draw a graph, and especially carefully reflect it convex/concave! Well, you probably realized a long time ago that the x-axis is horizontal asymptote, and it is absolutely forbidden to “climb” behind it! When filing a solution electronically, it’s easy to create a graph in Excel, and unexpectedly for myself, I even recorded a short video on this topic. But first, let's talk about how the shape of the normal curve changes depending on the values of and. When increasing or decreasing "a" (with constant “sigma”) the graph retains its shape and moves right/left respectively. So, for example, when the function takes the form In case of change of "sigma" (with constant “a”), the graph “stays the same” but changes shape. When enlarged, it becomes lower and elongated, like an octopus stretching its tentacles. And, conversely, when decreasing the graph becomes narrower and taller- it turns out to be a “surprised octopus”. Yes, when decrease“sigma” twice: the previous graph narrows and stretches up twice: A normal distribution with a unit sigma value is called normalized, and if it is also centered(our case), then such a distribution is called standard. It has an even simpler density function, which has already been found in Laplace's local theorem: Well, now let's watch the movie:
Yes, absolutely right - somehow undeservedly it remained in the shadows probability distribution function. Let's remember her definition: Inside the integral, a different letter is usually used so that there are no “overlaps” with the notation, because here each value is associated with improper integral Almost all values cannot be calculated accurately, but as we have just seen, with modern computing power this is not difficult. So, for the function =NORMSDIST(z) One, two - and you're done: Now let’s remember one of the key tasks of the topic, namely, find out how to find the probability that a normal random variable will take the value from the interval. Geometrically, this probability is equal to area between the normal curve and the x-axis in the corresponding section: ! Also remembers
, What Here you can use Excel again, but there are a couple of significant “buts”: firstly, it is not always at hand, and secondly, “ready-made” values will most likely raise questions from the teacher. Why? I have talked about this many times before: at one time (and not very long ago) a regular calculator was a luxury, and the “manual” method of solving the problem in question is still preserved in educational literature. Its essence is to standardize values “alpha” and “beta”, that is, reduce the solution to the standard distribution: Note
: the function is easy to obtain from the general case Why is this necessary? The fact is that the values were meticulously calculated by our ancestors and compiled into a special table, which is in many books on terwer. But even more often there is a table of values, which we have already dealt with in Laplace's integral theorem: If we have at our disposal a table of values of the Laplace function I remind you that Answer is required to be given as a percentage, so the calculated probability must be multiplied by 100 and the result provided with a meaningful comment: – with a flight from 5 to 70 m, approximately 15.87% of shells will fall We train on our own: Example 3 The diameter of factory-made bearings is a random variable, normally distributed with a mathematical expectation of 1.5 cm and a standard deviation of 0.04 cm. Find the probability that the size of a randomly selected bearing ranges from 1.4 to 1.6 cm. In the sample solution and below, I will use the Laplace function as the most common option. By the way, note that according to the wording, the ends of the interval can be included in the consideration here. However, this is not critical. And already in this example we encountered a special case - when the interval is symmetrical with respect to the mathematical expectation. In such a situation, it can be written in the form and, using the oddity of the Laplace function, simplify the working formula: It’s good that the solution fits in one line :) The result of this task turned out to be close to unity, but I would like even greater reliability - namely, to find out the boundaries within which the diameter is located almost everyone bearings. Is there any criterion for this? Exists! The question posed is answered by the so-called Its essence is that practically reliable
is the fact that a normally distributed random variable will take a value from the interval Indeed, the probability of deviation from the expected value is less than: In terms of bearings, these are 9973 pieces with a diameter from 1.38 to 1.62 cm and only 27 “substandard” copies. In practical research, the three sigma rule is usually applied in the opposite direction: if statistically It was found that almost all values random variable under study fall within an interval of 6 standard deviations, then there are compelling reasons to believe that this value is distributed according to a normal law. Verification is carried out using theory statistical hypotheses. We continue to solve the harsh Soviet problems: Example 4 The random value of the weighing error is distributed according to the normal law with zero mathematical expectation and a standard deviation of 3 grams. Find the probability that the next weighing will be carried out with an error not exceeding 5 grams in absolute value. Solution very simple. By condition, we immediately note that at the next weighing (something or someone) we will almost 100% get the result with an accuracy of 9 grams. But the problem involves a narrower deviation and according to the formula Answer: The solved problem is fundamentally different from a seemingly similar one. Example 3 lesson about uniform distribution. There was an error rounding measurement results, here we are talking about the random error of the measurements themselves. Such errors arise due to the technical characteristics of the device itself. (the range of acceptable errors is usually indicated in his passport), and also through the fault of the experimenter - when we, for example, “by eye” take readings from the needle of the same scales. Among others, there are also so-called systematic measurement errors. It's already non-random errors that occur due to incorrect setup or operation of the device. For example, unregulated floor scales can steadily “add” kilograms, and the seller systematically weighs down customers. Or it can be calculated not systematically. However, in any case, such an error will not be random, and its expectation is nonzero. …I’m urgently developing a sales training course =) Let’s solve the inverse problem ourselves: Example 5 The diameter of the roller is a random normally distributed random variable, its standard deviation is equal to mm. Find the length of the interval, symmetrical with respect to the mathematical expectation, into which the length of the roller diameter is likely to fall. Point 5* design layout to help. Please note that the mathematical expectation is not known here, but this does not in the least prevent us from solving the problem. And an exam task that I highly recommend to reinforce the material: Example 6 A normally distributed random variable is specified by its parameters (mathematical expectation) and (standard deviation). Required: a) write down the probability density and schematically depict its graph; Such problems are offered everywhere, and over the years of practice I have solved hundreds and hundreds of them. Be sure to practice drawing a drawing by hand and using paper tables;) Well, I’ll look at an example of increased complexity: Example 7 The probability distribution density of a random variable has the form Solution: First of all, let us note that the condition does not say anything about the nature of the random variable. The presence of an exponent in itself does not mean anything: it may turn out, for example, indicative or even arbitrary continuous distribution. And therefore the “normality” of the distribution still needs to be justified: Since the function Here we go. For this select a complete square and organize three-story fraction: Thus: Now let's find the value of the parameter. Since the normal distribution multiplier has the form and , then: Let's build a density graph: They say that CB X has uniform distribution in the area from a to b, if its density f(x) in this area is constant, that is For example, a measurement of some quantity is made using a device with rough divisions; the nearest integer is taken as an approximate value of the measured quantity. SV X - the measurement error is distributed uniformly over the area, since none of the values of the random variable is in any way preferable to the others. Exponential is the probability distribution of a continuous random variable, which is described by the density where is a constant positive value. An example of a continuous random variable distributed according to an exponential law is the time between the occurrences of two consecutive events of the simplest flow. Often the duration of failure-free operation of elements has an exponential distribution, the distribution function of which — failure rate (average number of failures per unit of time). Normal Law distribution (sometimes called Gauss's law) plays an extremely important role in probability theory and occupies a special position among other laws of distribution. The distribution density of the normal law has the form where m is the mathematical expectation, — standard deviation X. The probability that a normally distributed SV X will take a value belonging to the interval is calculated by the formula: where Ф(X) - Laplace function. Its values are determined from the table in the appendix of the textbook on probability theory. The probability that the deviation of a normally distributed random variable X from its mathematical expectation in absolute value is less than a given positive number is calculated using the formula EXAMPLE 13.2.41. The value of one division of the ammeter scale is 0.1 A. Readings are rounded to the nearest whole division. Find the probability that during the reading an error will be made that exceeds 0.02 A. Solution. The rounding error can be considered as CB X, which is distributed evenly in the interval between two adjacent divisions. Uniform distribution density , where (b-a) is the length of the interval that contains the possible values of X. In the problem under consideration, this length is 0.1. That's why The reading error will exceed 0.02 if it is in the interval (0.02; 0.08). According to the formula EXAMPLE 13.2.42. The duration of failure-free operation of an element has an exponential distribution. Find the probability that over a period of hours: a) the element fails; b) the element will not fail. Solution. a) The function determines the probability of failure of an element over a time duration t, therefore, by substituting , we obtain the probability of failure: . b) The events “the element will fail” and “the element will not fail” are opposite, so the probability that the element will not fail is . EXAMPLE 13.2.43. The random variable X is normally distributed with parameters . Find the probability that SV X will deviate from its mathematical expectation m by more than . This probability is very small, that is, such an event can be considered almost impossible (you can be wrong in about three cases out of 1000). This is the “three sigma rule”: if a random variable is normally distributed, then the absolute value of its deviation from the mathematical expectation does not exceed three times the standard deviation. EXAMPLE 13.2.44. The mathematical expectation and standard deviation of a normally distributed random variable are respectively equal to 10 and 2. Find the probability that, as a result of the test, X will take a value contained in the interval (12, 14). Solution: For a normally distributed quantity Substituting , we get We find from the table. The required probability. Solve problems using probability formulas for continuous random variables and their characteristics 3.2.9.1. Find the mathematical expectation, variance and standard deviation of a random variable X distributed uniformly in the interval (a,b). Rep.: 3.2.9.2. Subway trains run regularly at intervals of 2 minutes. A passenger enters the platform at a random time. Find the distribution density of SV T - the time during which he will have to wait for the train; Rep.: 3.2.9.3. The minute hand of an electric clock jumps at the end of each minute. Find the probability that at a given instant the clock will show a time that differs from the true time by no more than 20 s. Rep.:2/3 3.2.9.4. The random variable X is distributed uniformly over the area (a,b). Find the probability that as a result of the experiment it will deviate from its mathematical expectation by more than . Rep.:0 3.2.9.5. Random variables X and Y are independent and distributed uniformly: X in the interval (a,b), Y in the interval (c,d). Find the mathematical expectation of the product XY. Rep.: 3.2.9.6. Find the mathematical expectation, variance and standard deviation of an exponentially distributed random variable. Rep.: 3.2.9.7. Write the density and distribution function of the exponential law if the parameter . Rep.: 3.2.9.8. The random variable has an exponential distribution with parameter . Find Rep.:0,233 3.2.9.9. The failure-free operation time of an element is distributed according to the exponential law, where t is time, hours. Find the probability that the element will operate without failure for 100 hours. Rep.:0,37 3.2.9.10. Test three elements that operate independently of one another. The duration of failure-free operation of the elements is distributed according to the exponential law: for the first element Rep.: a)0.292; b)0.466; c)0.19 3.2.9.11. Prove that if a continuous random variable is distributed according to the exponential law, then the probability that X will take a value less than the mathematical expectation M(X) does not depend on the value of the parameter; b) find the probability that X > M(X). Rep.: 3.2.9.12. The mathematical expectation and standard deviation of a normally distributed random variable are respectively equal to 20 and 5. Find the probability that as a result of the test X will take a value contained in the interval (15; 25). Rep.: 0,6826 3.2.9.13. A substance is weighed without systematic errors. Random weighing errors are subject to the normal law with a standard deviation r. Find the probability that a) weighing will be carried out with an error not exceeding 10 r in absolute value; b) out of three independent weighings, the error of at least one will not exceed 4g in absolute value. Rep.: 3.2.9.14. The random variable X is normally distributed with mathematical expectation and standard deviation. Find the interval, symmetrical with respect to the mathematical expectation, into which, with a probability of 0.9973, the value X will fall as a result of the test. Rep.:(-5,25) 3.2.9.15. The plant produces balls for bearings, the nominal diameter of which is 10 mm, and the actual diameter is random and distributed according to the normal law with mm and mm. During inspection, all balls that do not pass through a round hole with a diameter of 10.7 mm and all that pass through a round hole with a diameter of 9.3 mm are rejected. Find the percentage of balls that will be rejected. Rep.:8,02% 3.2.9.16. The machine stamps parts. The length of the part X is controlled, which is distributed normally with a design length (mathematical expectation) equal to 50 mm. In fact, the length of the manufactured parts is no less than 32 and no more than 68 mm. Find the probability that the length of a randomly taken part: a) is greater than 55 mm; b) less than 40 mm. Hint: From equality Rep.:a)0.0823; b)0.0027 3.2.9.17. Boxes of chocolate are packed automatically; their average weight is 1.06 kg. Find the variance if 5% of the boxes have a mass less than 1 kg. It is assumed that the mass of the boxes is distributed according to the normal law. Rep.:0,00133 3.2.9.18. A bomber flying along the bridge, which is 30 m long and 8 m wide, dropped bombs. The random variables X and Y (the distance from the vertical and horizontal axes of symmetry of the bridge to the place where the bomb fell) are independent and normally distributed with standard deviations equal to 6 and 4 m, respectively, and mathematical expectations equal to zero. Find: a) the probability of one thrown bomb hitting the bridge; b) the probability of destruction of the bridge if two bombs are dropped, and it is known that one hit is enough to destroy the bridge. Rep.: 3.2.9.19. In a normally distributed population, 11% of X values are less than 0.5 and 8% of X values are greater than 5.8. Find the parameters of m and this distribution. > To problems 41-50.
looks like 
.
,
,
,
,
.
. Substituting the probability values into this formula, we get 
,
,
,
,
.
Normal probability distribution law
Here is a basic view density functions normal probability distribution, and I welcome you to this interesting lesson.
and our graph “moves” 3 units to the left - exactly to the origin of coordinates: 
A normally distributed quantity with zero mathematical expectation received a completely natural name - centered; its density function 
– even, and the graph is symmetrical about the ordinate.
Everything is in full accordance with geometric transformations of graphs.
. The standard distribution has found wide application in practice, and very soon we will finally understand its purpose.
– the probability that a random variable will take a value LESS than the variable that “runs through” all real values to “plus” infinity.
, which is equal to some number from the interval .
standard distribution, the corresponding Excel function generally contains one argument:
The drawing clearly shows the implementation of all distribution function properties, and from the technical nuances here you should pay attention to horizontal asymptotes and the inflection point.
but every time I try to get an approximate value 
is unreasonable, and therefore it is more rational to use "easy" formula:
.
using linear replacements. Then also:
and from the replacement carried out the formula follows: 
transition from the values of an arbitrary distribution to the corresponding values of a standard distribution.
, then we solve through it:
Fractional values are traditionally rounded to 4 decimal places, as is done in the standard table. And for control there is Point 5 layout.
, and to avoid confusion always control, a table of WHAT function is in front of your eyes.
The delta parameter is called deviation from the mathematical expectation, and the double inequality can be “packaged” using module:
– the probability that the value of a random variable will deviate from the mathematical expectation by less than .
– the probability that the diameter of a randomly taken bearing differs from 1.5 cm by no more than 0.1 cm.three sigma rule
.
or 99.73%
:
– the probability that the next weighing will be carried out with an error not exceeding 5 grams.
b) find the probability that it will take a value from the interval 
;
c) find the probability that the absolute value will deviate from no more than ;
d) using the “three sigma” rule, find the values of the random variable.
. Find, mathematical expectation, dispersion, distribution function, build density graphs and distribution functions, find.
determined at any real value, and it can be reduced to the form 
, then the random variable is distributed according to the normal law.
Be sure to perform a check, returning the indicator to its original form:
, which is what we wanted to see.
- By rule of operations with powers"pinch off" And here you can immediately write down the obvious numerical characteristics: 
, from where we express and substitute into our function: 
, after which we will once again go through the recording with our eyes and make sure that the resulting function has the form 
.
and distribution function graph 
:
If you don’t have Excel or even a regular calculator at hand, then the last graph can be easily constructed manually! At the point the distribution function takes the value 
and here it is
.
determines the probability of element failure over a time duration t.
,
,
.EXAMPLES OF SOLVING PROBLEMS
. So, 
.
we have 
.Examples and tasks for independent solution
. Find the probability that you will have to wait no more than half a minute.
, 
.
; for the second 
; for the third element 
. Find the probability that in the time interval (0; 5) hours: a) only one element will fail; b) only two elements; c) all three elements.
beforehand find .
Examples of problem solving >