A1*. ELEMENTS OF THE THEORY OF PROBABILITIES

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1 A*. ELEMENTS OF THE THEORY OF PROBABILITIES As it was shown in fame of hapte 5 Themodnamics its main elements: the state equations cannot be obtained onl b stating fom the themodnamic notions the deivation of these equations being possible b means of: a some epeiments b the theoetical notions of the Statistical Phsics o at least of its embo theo: the kinetic theo of gases elaboated b Mawell and Boltzmann. Taking into account that the main quantit specific to both the: i Statistical Phsics to the: ii Quantum Phsics and also to the: iii Laboato Data Pocessing is the pobabilit the main goal of this Appendi is to define and stud the main popeties of pobabilities. A.. DEFINITIONS AND PROPERTIES OF THE APPEARANE FREQUENIES OF THE PHYSIAL EVENTS Two phsical events E and E ae called incompatible if the concomitant simultaneous appeaance of these events is impossible: E E Ø. One event that poduces alwas is named a sue event smbol Ω. When the union 3 of a cetain numbe of events: E E... E n is equivalent to the sue event: U n E i Ω the set {E E... E n } is called a complete i collective of phsical events. E.g. the events consisting in the obtainment b the same student of the final gade 4 and of the final gade 5 in fame of the same unique eamination ae incompatible. E.g. the esult of a finished eamination has to be concetized b a gade o at least b a qualitative decision accepted o ejected. 3 The smbols and of the intesection and union opeations of sets wee intoduced fistl b the Piemontese mathematician Giuseppe Peano in alcolo geometico secundo l Ausdehnungslehe di H. Gassmann ajai vol. 888 p. 98.

2 396 PHYSIS FOR ENGINEERING STUDENTS Fo a we have: f δ a d f dh a + f dh a + a a+ f dh a a a+ whee is a ve small positive quantit 0. Taking into account that dh - a 0 fo a U a + and that due to the continuit of the function f we have: f fa fo a a + we obtain: f a δ a d f a dh a f a if a. A.7 The snthesis of the above obtained esults is epessed b the socalled sifting popet of the Diac s delta function: 0 fo a U f δ a d A.7 f a if : a. Fo f onstant 0 - and one obtains: δ a d δ a d and finall the nomalization popet of the Diac s delta function: δ a d. A.8 Solved Poblems A.*. Stating fom the epession A3.4. of the pobabilit densit coesponding to the nomal N-dimensional distibution find the epession of the pobabilit densit of the nomal -dimensional distibution n. Solution. Let: i i i i A.9 be the vaiance of the vaiable i and: i j ij i i j j A.0 the covaiance of vaiables i and j.

3 Appendices 397 In the bi-dimensional case we have: hence: det whee: A. is the coelation coefficient of vaiables and. Because: det one obtains: T hence: + T. Finall one finds: [ + π ep n A. whee: and A.3 ae the educed eos deviations of vaiables and. A.*. The pobabilit densit coesponding to the so-called nomal onedimensional Gauss distibution epatition is given b the epession: [ ep b a. Deive the epessions of: a the theoetical aveage value of the vaiable b the vaiance: of the vaiable c the nomalization constant stating fom the vaiance of the vaiable.

4 398 PHYSIS FOR ENGINEERING STUDENTS Solution: a Stating fom the definition A. of the aveage value one obtains: d ep [ a b d [ a b d b + b. b ep d Taking into consideation the nomalization condition of the pobabilit densit: d one finds that: [ a b ep b + d b ep[ a b b. a a d b b b ep [ a b [ a b ep b d a. Using the epession of the integation b pats : udv uv vdu one obtains: b ep[ a b + ep[ a b d d a a a a c Stating fom the nomalization condition of the pobabilit densit: d one finds: ep[ a b d e X dx whee: a X a b. Let: I e X dx e Y dy whee Y is a vaiable independent on X. onside the space of vaiables X and Y chosen as othogonal coodinates fig. A.4.

5 Appendices 399 epession that coesponds [see e.g. A.5 to a discete distibution with the pobabilities p i i... n. b One finds similal that - in this case - the nomalization condition coesponding to the continuous distibutions: n n n d piδ i d pi δ i d pi d i i i coincides [see e.g. the elation A.4 with that coesponding to the discete distibutions. SPEIAL REFERENES. W. T. Eadie D. Dijad F. E. James M. Roos B. Sadoulet Statistical Methods in Epeimental Phsics Noth-Holland Publ. omp. Amstedam-New Yok-Ofod 98.. a W. Ledeman chief edit. Handbook of Applicable Mathematics vol. II Pobabilit John Wile & Sons hicheste -New Yok 990; b ibid. vol. VI Statistics P. W. M. John Statistical Methods in Engineeing and Qualit Assuance John Wile & Sons New Yok hicheste John Wile & Sons M. R. Spiegel J. J. Schille R. A. Sinivasan Schaum s Outline of Pobabilit and Statistics nd edition McGaw Hill G. Gimmett D. Stizake Pobabilit and Random Pocesses 3 d edition Ofod Univesit Pess Ofod UK S. Ross A fist couse in Pobabilit 7 th edition Peason Intenational Edition 005.