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1 Engneeng Mahemacs 05 SUBJEC NAME : Pobably & Random Pocess SUBJEC CODE : MA645 MAERIAL NAME : Fomula Maeal MAERIAL CODE : JM08AM007 REGULAION : R03 UPDAED ON : Febuay 05 (Scan he above QR code fo he dec download of hs maeal) Name of he Suden: Banch: UNI-I (RANDOM VARIABLES) ) Dscee andom vaable: A andom vaable whose se of possble values s ehe fne o counably nfne s called dscee andom vaable Eg: () Le epesen he sum of he numbes on he dce, when wo dce ae hown In hs case he andom vaable akes he values, 3, 4, 5, 6, 7, 8, 9, 0, and So s a dscee andom vaable () Numbe of ansmed bs eceved n eo ) Connuous andom vaable: A andom vaable s sad o be connuous f akes all possble values beween cean lms Eg: he lengh of me dung whch a vacuum ube nsalled n a ccu funcons s a connuous andom vaable, numbe of scaches on a suface, popoon of defecve pas among 000 esed, numbe of ansmed n eo 3) SlNo Dscee andom vaable p ( ) f ( ) d F( ) P 3 E Mean p( ) 4 E Connuous andom vaable F( ) P f ( ) d Mean E f ( ) d p( ) E f ( ) d Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page

2 Engneeng Mahemacs 05 5 Va E E Va E E 6 Momen = E p 7 MGF M E e e p ( ) 4) E a b ae b 5) Va a b a Va 6) Va a by a Va b Va Y 7) Sandad Devaon Va 8) f ( ) F( ) 9) p( a) p( a) Momen = E f ( ) d MGF ( ) M E e e f d p A B 0) p A / B, pb 0 p B ) If A and B ae ndependen, hen p A B p A pb ) s Momen abou ogn = E = nd Momen abou ogn = E M M 0 = 0 (Mean) he co-effcen of = E! (h Momen abou he ogn) 3) Lmaon of MGF: ) A andom vaable may have no momens alhough s mgf ess ) A andom vaable can have s mgf and some o all momens, ye he mgf does no geneae he momens ) A andom vaable can have all o some momens, bu mgf does no es ecep pehaps a one pon 4) Popees of MGF: b ) If Y = a + b, hen MY e M a ) M M c c, whee c s consan ) If and Y ae wo ndependen andom vaables hen M Y M MY 5) PDF, MGF, Mean and Vaance of all he dsbuons: Sl Dsbuo PDF ( P( ) No n Bnomal n ) MGF Mean Vaance nc p q n q pe np npq Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page

3 Engneeng Mahemacs 05 Posson 3 Geomec 4 Unfom 5 Eponenal e e e! q p (o) q p pe q qe p p, a b f( ) b a 0, ohewse e, 0, 0 f( ) 0, ohewse e f ( ), 0, 0 ( ) 6 Gamma 7 Nomal f ( ) e b a e e a b ( b a) ( ) e ( b a) 6) Memoyless popey of eponenal dsbuon P S / S P d 7) Funcon of andom vaable: fy( y) f ( ) dy UNI-II (RANDOM VARIABLES) ) pj (Dscee andom vaable) j f (, y) ddy (Connuous andom vaable) P, y P( y) ) Condonal pobably funcon gven Y P / Y y P, y P ( ) Condonal pobably funcon Y gven PY y / P a / Y b P a, Y b P( Y b) 3) Condonal densy funcon of gven Y, f (, y) f ( / y) f( y) Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 3

4 Engneeng Mahemacs 05 Condonal densy funcon of Y gven, f (, y) f ( y / ) f( ) 4) If and Y ae ndependen andom vaables hen f (, y) f ( ) f ( y) (fo connuous andom vaable), P Y y P P Y y (fo dscee andom vaable) d b, (, ) 5) Jon pobably densy funcon P a b c Y d f y ddy b a P a, Y b f (, y) ddy 6) Magnal densy funcon of, f ( ) f ( ) f (, y) dy Magnal densy funcon of Y, f ( y) fy ( y) f (, y) d 7) P( Y ) P( Y ) 0 0 c a 8) Coelaon co effcen (Dscee): Cov(, Y ) (, y) Y Cov(, Y ) Y Y, n n, Y Y Y n Cov(, Y ) 9) Coelaon co effcen (Connuous): (, y) Cov(, Y ) E Y E E Y, Va( ), Va( Y ) 0) If and Y ae uncoelaed andom vaables, hen Cov(, Y ) 0 ) E f ( ) d, E Y yf ( y) dy, E, Y yf (, y) ddy Y Y ) Regesson fo Dscee andom vaable: Regesson lne on Y s b y y, y y y y y b y Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 4

5 Engneeng Mahemacs 05 Regesson lne Y on s y y b y y y, b y Coelaon hough he egesson, b b Noe: (, y) (, y) Y Y 3) Regesson fo Connuous andom vaable: Regesson lne on Y s E( ) b y E( y) y, by y Regesson lne Y on s y E( y) b E( ), y y by Regesson cuve on Y s / / E y f y d Regesson cuve Y on s / / y E y y f y dy 4) ansfomaon Random Vaables: d fy( y) f ( ) (One dmensonal andom vaable) dy u v fuv ( u, v) f Y (, y) (wo dmensonal andom vaable) y y u v UNI-III (MARKOV PROCESSES AND MARKOV CHAINS) ) Random Pocess: A andom pocess s a collecon of andom vaables {(s,)} ha ae funcons of a eal vaable, namely me whee s Є S and Є ) Classfcaon of Random Pocesses: We can classfy he andom pocess accodng o he chaacescs of me and he andom vaable We shall consde only fou cases based on and havng values n he anges - < < and - < < Connuous andom pocess Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 5

6 Engneeng Mahemacs 05 Connuous andom sequence Dscee andom pocess Dscee andom sequence Connuous andom pocess: If and ae connuous, hen we call (), a Connuous Random Pocess Eample: If () epesens he mamum empeaue a a place n he neval (0,), {()} s a Connuous Random Pocess Connuous Random Sequence: A andom pocess fo whch s connuous bu me akes only dscee values s called a Connuous Random Sequence Eample: If n epesens he empeaue a he end of he nh hou of a day, hen {n, n 4} s a Connuous Random Sequence Dscee Random Pocess: If assumes only dscee values and s connuous, hen we call such andom pocess {()} as Dscee Random Pocess Eample: If () epesens he numbe of elephone calls eceved n he neval (0,) he {()} s a dscee andom pocess snce S = {0,,,3, } Dscee Random Sequence: A andom pocess n whch boh he andom vaable and me ae dscee s called Dscee Random Sequence Eample: If n epesens he oucome of he nh oss of a fa de, he {n : n } s a dscee andom sequence Snce = {,,3, } and S = {,,3,4,5,6} 3) Condon fo Saonay Pocess: E ( ) Consan, If he pocess s no saonay hen s called evoluonay Va ( ) consan 4) Wde Sense Saonay (o) Weak Sense Saonay (o) Covaance Saonay: A andom pocess s sad o be WSS o Covaance Saonay f sasfes he followng condons E ( ) consan ) he mean of he pocess s consan (e) ) Auo coelaon funcon depends only on (e) R ( ) E ( ) ( ) 5) me aveage: he me aveage of a andom pocess () s defned as ( ) d Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 6

7 Engneeng Mahemacs 05 If he neval s 0,, hen he me aveage s ( ) d 0 6) Egodc Pocess: A andom pocess () s called egodc f all s ensemble aveages ae nechangeable wh he coespondng me aveage 7) Mean egodc: Le () E () hen () s sad o be mean egodc f be a andom pocess wh mean E () L Noe: L va 0 (by mean egodc heoem) and me aveage, as (e) 8) Coelaon egodc pocess: he saonay pocess () s sad o be coelaon egodc f he pocess Y () s mean egodc whee Y( ) ( ) ( ) E Y L Y Whee Y s he me aveage of Y () 9) Auo covaance funcon: C ( ) R ( ) E ( ) E ( ) 0) Mean and vaance of me aveage: Mean: ( ) Vaance: 0 (e) E E d Va R ( ) C ( ) d () ) Makov pocess: A andom pocess n whch he fuue value depends only on he pesen value and no on he pas values, s called a makov pocess I s symbolcally epesened by P ( n) n / ( n) n, ( n ) n ( 0) 0 P ( n) n/ ( n) n Whee 0 n n ) Makov Chan: If fo all n, P n an / n an, n an, 0 a0 P n an / n an hen he pocess n, n 0,,, s called he makov chan Whee a0, a, a, a n, ae called he saes of he makov chan 3) anson Pobably Ma (pm): When he Makov Chan s homogenous, he one sep anson pobably s denoed by Pj he ma P = {Pj} s called anson pobably ma Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 7

8 Engneeng Mahemacs 05 4) Chapman Kolmogoov heoem: If P s he pm of a homogeneous Makov chan, hen he n sep pm P (n) s equal o P n (e) P ( n) j j n P 5) Makov Chan popey: If,, 3, hen P and 3 6) Posson pocess: If () epesens he numbe of occuences of a cean even n (0, ),hen he dscee andom pocess () s called he Posson pocess, povded he followng posulaes ae sasfed () P occuence n (, ) O () P0 occuence n (, ) O () P o moe occuences n (, ) O (v) () s ndependen of he numbe of occuences of he even n any neval 7) Pobably law of Posson pocess: Mean E (), e P ( ), 0,,,! E () Va (), UNI-IV (CORRELAION AND SPECRAL DENSIY) R S RY SY - Auo coelaon funcon - Powe specal densy (o) Specal densy - Coss coelaon funcon - Coss powe specal densy ) Auo coelaon o Powe specal densy (specal densy): S R e d ) Powe specal densy o Auo coelaon: R S e d Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 8

9 Engneeng Mahemacs 05 3) Condon fo () and ( ) ae uncoelaed andom pocess s C ( ) R ( ) E ( ) E ( ) 0 4) Coss powe specum o Coss coelaon: RY SY e d UNI-V (LINEAR SYSEMS WIH RANDOM INPUS) ) Lnea sysem: f s called a lnea sysem f sasfes f a ( ) a( ) a f ( ) a f ( ) ) me nvaan sysem: Le Y( ) f ( ) If Y( h) f ( h) nvaan sysem 3) Relaon beween npu () and oupu Y (): Y( ) h( u) ( u) du Whee hu ( ) sysem weghng funcon hen f s called a me 4) Relaon beween powe specum of () and oupu Y (): S ( ) S ( ) H( ) YY j If H( ) s no gven use he followng fomula H( ) e h( ) d 5) Conou negal: m e ma e (One of he esul) a a 6) F a a e a (fom he Foue ansfom) ---- All he Bes ---- Pepaed by CGanesan, MSc, MPhl, (Ph: ) Page 9

I-POLYA PROCESS AND APPLICATIONS Leda D. Minkova

I-POLYA PROCESS AND APPLICATIONS Leda D. Minkova The XIII Inenaonal Confeence Appled Sochasc Models and Daa Analyss (ASMDA-009) Jne 30-Jly 3, 009, Vlns, LITHUANIA ISBN 978-9955-8-463-5 L Sakalaskas, C Skadas and E K Zavadskas (Eds): ASMDA-009 Seleced