DISTRIBUTION LAW Okunev I.V.

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1 1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated mathematical laws better tha examiig cocrete practical cases. This examiatio shows that theoretical ad practical distributio laws are expressed i differet mathematical formulas. This ad may others is the topic of the article. A practical example from the mechaical egieerig extremely simplifies uderstadig the article. Key words: distributio law, mechaical egieerig, mathematical research. The MSC code: 62E15. Let's cosider the followig case from the mechaical egieerig. A desiger had made a drawig of a part, where he had idicated its omial dimesio ad the admissible deviatios of the dimesio. A worker has made parts accordig to the drawig. The parts have differet dimesios, however, all the dimesios are withi the specified toleraces. The smallest dimesio is equal k. There are k parts with this dimesio. The largest dimesio is equal m. There are m parts with this dimesio. As it is kow, the frequecy of a evet A (or the statistical probability of a evet А) i a give series of experimets is the ratio of the quatity of the experimets, i which the evet A appeared, to the total quatity of the experimets made [1]. If a evet A is the emergece of a part with a dimesio x, ad x is the quatity of the parts with the dimesio х, the the frequecy of the evet A for the dimesio х is equal x. The sum of all the values of the ratio x, whe х takes the values from k to x, is the itegral distributio law of the quatity х (i the work, for coveiece, «the distributio law»). It is deoted with F(х). The: F x = x, F k = k ad F m = =1. The geeral formula of distributio law should be established first for theoretical cases, i.e. whe the iterval betwee the adjacet values of the quatity x is a ifiitesimal dx, this iterval is the same for all x i the distributio iterval, ad, therefore, the quatity of the itervals dx is

2 2 equal dx. The: F x = x =df x It is obvious that the geeral formula of distributio law must be foud by a trasitio from the fuctio F(x) to some arbitrarily defied fuctio П(х). The fuctio F(x) ca ot be defied arbitrarily, because it is the sum of the values of the expressio df(x). The expressio df(x) ca ot be defied arbitrarily, because it is the ratio of the quatities x ad, which are ot coected by a mathematical relatioship. The, the trasitio from F(x) to П(х) must be carried out by meas of the elimiatio of the quatity. I order to elimiate the quatity, it is ecessary to fid the quotiet of the values of the expressio df(x) at two adjacet poits. This operatio results i the followig expressio: df x dx df x = df x d 2 F x df x =1 d 2 F x F ' ' x =1 df x F ' x The product of all the values of the quatity 1 F ' ' x F ' x dx, whe x dx takes the values from k to x, is equal df x df k, i.e. the quatity 1 F ' ' x F ' x dx is a cofactor i the fuctio df x df k. It is quite obvious that i the fuctio П(х) the quatity 1 F ' ' x F ' x dx must also be a cofactor. However, because this fuctio is essetially ew, the the multiplicatio is made ot over the variable x but over some other variable, the differetial of which is also equal dx, but the variable itself is idepedet of x. It is obvious that this variable is oe of the two limits of the distributio iterval, i.e. either k or m. The, the fuctio П(х) is the product of the idetical cofactors 1 F ' ' x F ' x dx, ad the quatity of these cofactors is equal dx, i.e.:

3 3 F ' ' x П х =[ 1 F ' x dx ] F ' ' x dx F ' x =e Whece, the geeral formula of distributio law for theoretical cases is the followig oe: F ' ' x F ' x =l П х, where: П(х) a arbitrarily defied fuctio; - the legth of the distributio iterval. While defiig the fuctio П(х), some volume of iformatio eters ito it. As a result of the itegratio, this iformatio eters ito the fuctio F(x) i a distorted form. Hece, at a iverse process, i.e. at a trasitio from the fuctio F(x) to the fuctio П(х), there will be a correctio of iformatio. It is quite obvious, that a ormal distributio takes place whe the coditios of the experimet are ideal. I this case, the fuctio П(х) cotais a miimal volume of iformatio ad is a liear depedece o х, i.e. П х =c d x, where c ad d are costats. However, uder such a coditio, the fuctio F(x) caot be expressed i the elemetary fuctios. If П(х) is equal 1 x ad all the values of х are close to 0, the the followig approximate equality is true: The: l П х = х d F ' x F ' x = x dx Let's itegrate this expressio for x betwee the limits from m to х: Whece: i.e. we came to the Gaussia law. l F ' x l F ' m = x 2 m 2 2 x 2 m 2 2 F ' x =F ' m e, Let's itroduce a fuctio T(x) that is coected with the fuctio F(x) by the followig relatioship:

4 4 F x = k 1 k T x =F k [1 F k ] T x It is sigificat that T k =0 ad T m =1. The fuctios F(x) ad T(x) are mootoically icreasig oes. I practice, the curves of these fuctios have a quatity of iflectio poits (Fig.1). As a result of this, the values of the fuctios F ' x ad F ' ' x F ' x e hut about some average values of these fuctios. I this case, the geeral formula of distributio law for theoretical cases is ot applicable. I order to trasit to a formula that is applicable i practice it is ecessary to replace i the specified formula the fuctios F ' x ad F ' ' x F ' x e with their average values. Let Б(x) is the fuctio that represets the average values of the fuctio F ' x i the iterval [k;x]. Because the fuctio F ' x dx=df x is a summad of the fuctio F(x), the these values are foud as arithmetic meas, i.e. from the fuctio F(x) is subtracted its value at the poit k, ad the result is divided by x k :

5 5 Б х = F x F k x k The value of the fuctio Б(x) at the poit k is a idetermiate oe: Б k = 0 0 The value of this fuctio at the poit m is a determiate oe: 1 F k Б m = Hece, it is ecessary to fid the average values of the fuctio Б ' x Б x e i the iterval [x;m]. Because fuctio e [ Б ' x Б x ] dx is a cofactor of the fuctio Б(x), the these values are foud as geometric meas, i.e. the fuctio Б(x) is divided by its value at the poit m, ad the root is take of the result. The fuctio obtaied is equal П х Whece: Fially: it i detail. [ Б х Б m ] 1 : 1 =П х 1 Б х [ F x F k ] = =П х Б m x k [1 F k ] F x =F k [1 F k ] x k П х This is the geeral formula of distributio law for practical cases. The: T x = x k П х This formula is irreplaceable i practical research, therefore, we examie The formula for the distributio desity T ' x is the followig oe: [ 1 l П х T ' x =T x x k П ' x П x ]

6 6 At the poits k ad m this fuctio takes the followig values: T ' k = П k ; T ' m =1 l П m I order to carry out research, it is ecessary to calculate the exact values of the fuctio Т(х) at all the poits for which there are experimetal data. Some researchers, aspirig to make the plot of the fuctio Т(х) smoother, chage the values of the fuctio Т(х) a little, which is ot recommeded to do, sice, as a result, there ca be a essetial error i research. The, o the basis of the values of the fuctio Т(х) available, the values of the fuctio П(х) are calculated by the followig formula: П x =[ x k T х ] As said above, if the coditios of the experimet are ideal, the the fuctio П(х) cotais a miimal volume of iformatio ad is a liear depedece o х. The simplest case of such a distributio is a straight lie T x = x k whe П x =1. At a excess over a miimal volume, the

7 7 depedece П(х) o х becomes oliear. If such a excess is ot sigificat, the the curve П(х) has o extreme. If such a excess is sigificat, the some extremes appear o the curve П(x). They ca sigal that: 1. The material of the blak has a defect i the distributio iterval. 2. The feeder of the machie tool has a defect i the positio whe the cuttig edge of the cutter is i the distributio iterval. 3. The tool which was used to measure the fiished parts has a defect i the distributio iterval. 4. The method of measurig the fiished parts has caused the biggest error i the distributio iterval, etc. A practical example from the mechaical egieerig. A turer has made a big quatity of parts "Plug". The plug s diameter specified o the drawig is 30h14 0 0,52 (mm). 46 parts are at radom chose from those parts. After measurig them it appeared that their diameters are i the limits from 29,575 mm to 29,68 mm. As i this case the dimesio of a fiished part is less tha the dimesio of a blak, the quatity х is the differece of the dimesio 30 mm ad the dimesio of a fiished part. The х is i the limits from 0,32 mm to 0,425 mm. The curve of distributio of the quatity х, Т(х), is show o Fig.1. The curve П(x) correspodig to it is show o Fig.2. O it there are 4 extremes. Two extremes at the begiig ad at the ed are explaied by the fact that the distributio iterval of the parts chose does ot coicide with the actual distributio iterval. Two other extremes are explaied by some iaccuracy i the method of measuremet. LIST OF REFERENCES [1] Vetsel, E.S. (1969). Теоriya veroyatostey (Probability theory), 4 th editio, "Nauka", Moscow, Russia.

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

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