The Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof

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1 MATEC Web of Cofeeces ICIEA (06) DOI: 0.05/mateccof/ The ea Pobablty Desty Fucto of Cotuous Radom Vaables the Real Numbe Feld ad Its Estece Poof Yya Che ad Ye Collee of Softwae, Taj Uvesty, Taj, Cha Collee of Mathematcal Scece, Captal Nomal Uvesty, Bej, Cha Abstact. Fo a cotuous adom vaable eal umbe feld, thee must be a dstbuto ad also a pobablty desty fucto of ths adom vaable. If thee s a ow fucto wth ths adom vaable as depedet vaable, ts mae s a smooth o pecewse smooth le, thee must be at least oe fucto that taes ths adom vaable as ts depedet vaable, these fuctos ae bouded o the mae of the fst fucto. Ay oe of these fuctos coduct le teal opeato to the le semet o ac leth of the ceta mae of the fst ow fucto s the cumulatve pobablty of ths cotuous adom vaable teval coespod to the secto of the mae fo le teal opeato. A eeal desato fo these fuctos ae lea pobablty desty fucto of cotuous adom vaables. Coduct le teal opeato to the lea pobablty desty fucto ad coduct teal opeato to the pobablty desty fucto have same esults of the cumulatve pobablty of cotuous adom vaable. By the way, e teato clud cuve teato. Accod to the uqueess of the pobablty, the estece ad the umbe of lea pobablty desty fucto ca be poved ad calculated. Itoducto Fo a cotuous adom vaable as X, X R, whch meet the codto X, obvously, thee s a X pobablty desty fucto f, meet the codto that f t dt F. F s the cumulatve pobablty fucto of X. Wth efeece to eamples of cuvlea teal [,], If the teal doma of X, whch equvalet to teval of the value of adom vaable X fom eal umbe as, chaes to the fucto mae ( o a pat of t ) of whch s a cotuous fucto o coutable pecewse fucto, thee should be aothe fucto h eal umbe feld, accod to the uqueess of pobablty, t meets huduf. The teato path of cotuous adom vaable X s a mov path of the pots,, composed of the value of X ad ts coespod value of the value, o the mov the decto of -as. h ae called the lea pobablty desty fucto of the cotuous adom vaable X the eal umbe feld. To ths ed, ts eed to defe the lea pobablty desty fucto, ad pove ts estece ad the umbe of the fucto eal umbe feld. Relevat defto Hee s a eed to ve defto ad popetes of lea teal, lea pobablty desty fucto.. ea teal.. Defto of ea teal Set X, X R, X s a cotuous adom vaable the eal umbe feld, X. Its pobablty desty X fucto s f, cumulatve pobablty fucto s F, set s a fucto of adom vaables wth X as depedet vaable, ts mae s smooth o pecewse smooth plae Oy, thee s aothe fucto h wth depedet vaable the same as X, h s bouded o the mae of, tecept a mae fom, ts edpots ae A, m m ad B,, us ulmted dvso, set a fte pot ae M, M,, M the, dvde to shot sectos, set the leth of the th secto s, s ay pot o the th secto, mae, also s The Authos, publshed by EDP Sceces. Ths s a ope access atcle dstbuted ude the tems of the Ceatve Commos Attbuto cese 4.0 (

2 MATEC Web of Cofeeces 600 (06) DOI: 0.05/mateccof/ ICIEA 06 poduct h,, s,,, poduct sums h,, s,,, f ma h, 0, lm h,,, ad also mae,, ths case, s s s 0 always est, the called ths lmt as a fucto h, ( ) wth do lea teal to o mae of. The mae of could be a staht le, could also be a cuve, could be bouded but also could be ubouded, but must meet the smooth o pecewse smooth... Classfcato of solv lea teal s bouded, set, If the mae of accod to the methods just ow, ma the lea teal as h, ds, thee ae: h, ds lm h, () s 0 If s ubouded, the two edpots of ae A, m m ad B,, f t s eed to equed out lea teal of, thee ae h, ds h, ds h, lm AB 0 () s If s ubouded, the, f t s eed to equed out lea teal of, set a fte pot ae M, M,, M, dvde to shot sectos,, set the leth of th secto s s, also, poduct h, s s ay pot o the th secto, mae,,,,, ths case, also mae h,, f h 0 h, lm s s ma, 0 s, s always est, the called ths lmt as a fucto h, ( ) wth do lea teal to o mae of. Stll ma the lea teal of as h, ds, thee ae:, lm h, h ds (3) s 0 I shot, f t s a eed to fd out the ete o a pat of mae of fo opeat lea teal, the method of fte dvso could be used to ealze to solve the lea teal calculato tadtoal.. ea pobablty desty fucto.. Defto of lea pobablty desty fucto Accod to the last secto, efeece to the defto ad popetes of pobablty desty fucto ad cumulatve pobablty fucto [3,4,5], h s a fucto of X s also a fucto of X, set F as the cumulatve pobablty fucto of X, hudu s opeat lea teal by -as decto alo the mae of, f hudu F (4) It s sad s path of lea teal of X, h s the lea desty fucto of X. Aothe ote that h,, h s ot a vaable of twodmesoal adom but a paamete of h... Popetes of lea pobablty desty fucto It s obvous that lea teal of the lea pobablty desty fucto s a chae fom of the cuve teal, ad the lea pobablty desty fucto s also a chae fom of pobablty desty fucto, so the lea pobablty desty fucto has the chaactestcs of both of them. Popety. If h s the lea pobablty desty fucto of a cotuous adom vaable X,the: h 0 (5) Popety. If h s the lea pobablty desty fucto of a cotuous adom vaable X, s the path of lea teal, epess the whole mae of o -as, the hudu (6) Popety 3. Set as, costats, h s the lea pobablty desty fucto of a cotuous adom vaable X, s the path of lea teal, f h, h, h uu, h uu, du h, h, h,the (7) uu du uu du Popety 4. h s the lea pobablty desty fucto of a cotuous adom vaable X, s the path of lea teal, set the whole mae of, ths case, mae o the -as s, f could be dvded to a fte umbe of shot sectos as,,,,,

3 MATEC Web of Cofeeces ICIEA (06) DOI: 0.05/mateccof/ , the: hudu hudu (8) Popety 5. h s the lea pobablty desty fucto of a cotuous adom vaable X, s the path of lea teal, set the whole mae of o the -as s, ths case, mae, f h t o the, the hudu t udu (9) h Popety 6. s the lea pobablty desty fucto of a cotuous adom vaable X, s the path of lea teal, f s the pobablty desty fucto of X, F s cumulatve pobablty of X,the h u du f t dt F (0) Popety 7 s the uqueess of the pobablty, fo a cotuous adom vaable, t s boud to be the oly estece of the cumulatve pobablty a ve eo, o matte what fom of epesso of pobablty. The follow s a poof of t. Poof: X s a cotuous adom vaable, ts pobablty desty fucto s f, ts cumulatve pobablty s F, lea pobablty desty fucto s, a b h, path of lea teal s epess the mae about a, b o the,f thee s a specfed teal eo whch le a b, b b f d a h d, set ab f d a P, set a b h d P,the Pa b P ad P thee ae two values, ad meawhle, the est of the teal eo of adom vaables satsfed h u du f t dt F, set the cumulatve pobablty of the est of teal eo of adom vaables as P, accod to the eulaty of f s,t s a eed to meet f t dt, but P P, obvously, P P ad s P P must have oe ot equal to, t s s usatsfed to the eulaty of pobablty desty fucto, theefoe, fo ay X, satsfed to h u du f t dt F. That s to say, o matte the cumulatve pobablty epess by the teal of pobablty desty fucto o lea teal of lea pobablty desty fucto, fo ay value eo of adom vaable X must be equal eveywhee. 3 Poof of estece of lea pobablty desty ad ts umbe Accod to the uqueess of the pobablty, t could pove the estece ad umbe of lea pobablty desty fucto. Thee ae two ds of stuatos, fst oe, f s ow, h s uow, h ; Secod oe, f s ow, h s uow, h h. The mea of f, ad ae the same as above secto. If to pove the estece of lea pobablty desty fucto, that s to pove the estece of h, t could be based o the equato of the pobablty le hudu f t dt F to solve h, f the equato has eal oots, t s poved the estece of h eal umbe feld, the umbe of eal oots s the umbe of h eal umbe feld. 3. Fst stuato Poof: The mea of f s ow, h s uow, h, f, ad h ae the same as above secto.to efeece the cotet of calculato method of cuve teal of ac leth mathematcal aalyss, t could povde that : hudu f t dt F u hu du F See the fst devatve o both sdes, the: h f h h f f h f f ad ae both eal fuctos, obvously, h ae also eal fuctos, theefoe, eal umbe feld, the estece of h s poved. The umbe of h s. 3. Secod stuato f s ow, h s uow, Poof: h, The mea of f, ad h ae the same as above secto.to efeece the cotet of calculato 3

4 MATEC Web of Cofeeces ICIEA (06) DOI: 0.05/mateccof/ method of cuve teal of ac leth mathematcal aalyss, t could povde that : hudu f t dt F u u du F See the fst devatve o both sdes, the: f f f f f Set, the, 0, substtute them to the equato, the : f f Set f C, substtute t to the equato, the : C C C 0 0 Solv hhe ode equato, as lo as see out, thus to see out h, wth the ad of MATAB, put code : syms ;=solve(*^6-^4-), t could et: = /6//(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/3) *6^(/)*(*(08*^+8+*3^(/)*(7*^+4)^(/)* )^(/3)*((08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)+4+*(08*^+8+*3^(/)*(7*^+4)^(/)* )^(/3)))^(/) -/6//(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/ 3)*6^(/)*(*(08*^+8+*3^(/)*(7*^+4)^(/) *)^(/3)*((08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)+4+*(08*^+8+*3^(/)*(7*^+4)^(/)* )^(/3)))^(/) /6//(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/3) *3^(/)*(*(08*^+8+*3^(/)*(7*^+4)^(/)* )^(/3)*(*3^(/)*(08*^+8+*3^(/)*(7*^+4) ^(/)*)^(/3)-(08*^+8+*3^(/)*(7*^+4)^(/ )*)^(/3)-4+4*(08*^+8+*3^(/)*(7*^+4)^ (/)*)^(/3)-4**3^(/)))^(/) -/6//(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/ 3)*3^(/)*(*(08*^+8+*3^(/)*(7*^+4)^(/) *)^(/3)*(*3^(/)*(08*^+8+*3^(/)*(7*^+ 4)^(/)*)^(/3)-(08*^+8+*3^(/)*(7*^+4)^ (/)*)^(/3)-4+4*(08*^+8+*3^(/)*(7*^+4) ^(/)*)^(/3)-4**3^(/)))^(/) /6//(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/3) *(-3**(08*^+8+*3^(/)*(7*^+4)^(/)*)^ (/3)*(*3^(/)*(08*^+8+*3^(/)*(7*^+4)^ (/)*)^(/3)+(08*^+8+*3^(/)*(7*^+4)^(/ )*)^(/3)+4-4*(08*^+8+*3^(/)*(7*^+4)^ (/)*)^(/3)-4**3^(/)))^(/) -/6//(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/ 3)*(-3**(08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)*(*3^(/)*(08*^+8+*3^(/)*(7*^+4)^ (/)*)^(/3)+(08*^+8+*3^(/)*(7*^+4)^(/ )*)^(/3)+4-4*(08*^+8+*3^(/)*(7*^+4)^ (/)*)^(/3)-4**3^(/)))^(/) Thee ae eal oots ad 4 maay oots. See out h eal umbe feld, see ecpocal ofeal oots of by MATAB, the: = *(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/3)*6 ^(/)/(*(08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)*((08*^+8+*3^(/)*(7*^+4)^(/)*)^ (/3)+4+*(08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)))^(/) -*(08*^+8+*3^(/)*(7*^+4)^(/)*)^(/3)* 6^(/)/(*(08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)*((08*^+8+*3^(/)*(7*^+4)^(/)*)^ (/3)+4+*(08*^+8+*3^(/)*(7*^+4)^(/)*) ^(/3)))^(/) Replace wth f, the t could et : h= f *(08* f ^+8+*3^(/)*(7* f ^+4)^(/)* f )^(/3)*6^(/)/( f *(08* f ^ +8+*3^(/)*(7* f ^+4)^(/)* f )^(/3)* ((08* f ^+8+*3^(/)*(7* f ^+4)^(/)* f )^(/3)+4+*(08* f ^+8+*3^(/)*(7* f ^+4)^(/)* f )^(/3)))^(/) h=- f *(08* f ^+8+*3^(/)*(7* f ^+4)^(/)* f )^(/3)*6^(/)/( f *(08* f ^ +8+*3^(/)*(7* f ^+4)^(/)* f )^(/3)* ((08* f ^+8+*3^(/)*(7*^+4)^(/)* f ) ^(/3)+4+*(08* f ^+8+*3^(/)*(7* f ^ +4)^(/)* f )^(/3)))^(/) 4

5 MATEC Web of Cofeeces ICIEA (06) DOI: 0.05/mateccof/ f s eal fuctos, obvously, h ae also eal fuctos, theefoe, eal umbe feld, the estece of h s poved. The umbe of h s. I shot, fo two ds of stuato above, thee ae 4 h eal umbe feld. 4 Bef summay ea pobablty desty fucto s a d of pobablty desty fucto, opeat lea teal fo lea pobablty desty fucto s aothe epesso of see cumulatve pobablty. It could accod to the ow specfed path to obta dffeet lea pobablty desty fucto. Refeeces. Zho, Fa p, A couse mathematcal aalyss, Hhe Educato Pess, Bej, (008): Depatmet of Appled Mathematcs of Toj Uvesty, Hhe mathematcs, Hhe Educato Pess, Bej, (00): Mao Shso, Pu Xaolo, Che Ym, A cocse o pobablty theoy ad mathmatcal statstcs, Hhe Educato Pess, Bej, (0): Mao Shso, v Xaol, Mathematcal Statstcs(Cha Rem Uvesty Pess, Bej, (0), Wu Xzh, Statstcs: fom data to coclusos, Cha Statstcs Pess, Bej, (0),