The Cumulive Disribuio Fucio (cd) ONE RANDOM VARIABLE cd is deied s he probbiliy o he eve { x}: F ( ) [ ] x P x x - Applies o discree s well s coiuous RV. Exmple: hree osses o coi x 8 3 x 8 8 F 3 3 7 x 3 8 8 8 8 3 3 x 3 8 8 8 8 The cd c be wrie compcly i erms o he ui sep ucio: x ux ( ) x 3 3 F u u( x ) u( x ) u( x 3) 8 8 8 8 Exmple: The Uiorm RV i he ui iervl [,]. x F x x x
Exmple: The wiig ime o cusomer xi sd is zero i he cusomer ids xi prked he sd, d uiormly disribued rdom legh o ime i he iervl [, ] (i hours) i o xi is oud upo rrivl. The probbiliy h xi is he sd whe he cusomer rrives is p. Fid he cd o. Use ol probbiliy heorem, F P[ x] P[ x id xi] p P[ x o xi]( p) x P[ x id xi] x x P[ x o xi] x x x bsic properies o he cd (i) F. (ii) lim F. x (iii) lim F. x x F p ( p) x x x (iv) F is odecresig ucio o x, h is, i b, F( ) F( b) (odecresig ucio). (v) F is coiuous rom he righ, h is, or h, F ( b) lim F ( b h) F ( b ). (vi) P[ b] F( b) F( ) (vii) P[ b] F( b) F( b ) (i F hs jump x = b). (viii) P[ x] F. The Three Types o Rdom Vribles h Discree rdom vribles: hve cd h is righ-coiuous, sircse ucio o x, wih jumps couble se o pois x, x, x,... F p ( x ) p ( x ) u( x x ) k k k xk x k Coiuous rdom vribles: is coiuous everywhere, d which, is suiciely smooh h i c be wrie s iegrl o some oegive ucio (x): x F dx P[ x] Rdom vrible o mixed ype: is rdom vrible wih cd h hs jumps o couble se o pois x, x, x,... bu h lso icreses coiuously over les oe iervl o vlues o x.
Cd: F pf ( p) F p, F : cd o discree RV; F : cd o co. RV The Probbiliy Desiy Fucio (pd) probbiliy desiy ucio o (pd) : df dx The pd represes he desiy o probbiliy he poi x: Properies: F( x h) F P[ x x h] F ( x h) F h h h (i) ( F is o-decresig ucio o x). b (ii) P[ x b] dx probbiliy o iervl [, b] x (iii) F ( ) d. (iv) dx. pd o Discree Rdom Vribles df p ( xk ) ( x xk ) dx k
Codiiol cd s d pd s codiiol cd o give C P[{ x} C] F ( x C) P[ C] PC [ ] codiiol pd o give C : df ( x C) ( x C) dx Exmple: The lieime o mchie hs coiuous cd F ( x ). Codiiol cd d pd give he eve C = { > } (mchie sill workig ime ) codiiol cd: P[{ x} { }] F ( x ) P[ ] x P[{ x} { }] P[ x] F F( ) x P[ ] P[ ] F ( ) F ( x ) F F( ) F ( ) x x codiiol pd: ( x ) x F ( ) cd o i erms o he codiiol cd s; priio o S: B, B,..., B F P[ x] P[ x B ] P[ B ] F ( x B ] P[ B ] i i i i i i ( x B ] P[ B ] i i i Exmple: A biry rsmissio sysem seds bi by rsmiig ( v) volge sigl, d bi by rsmiig (+v). Received sigl corruped by Gussi oise: Y = + N : he rsmied sigl, N is oise volge wih pd ( x ) N
Assume P[ ] = p = P[ ]. Fid he pd o Y. B = { is rsmied} B = { is rsmied}. F F ( x B ] P[ B ] F ( x B ] P[ B ] Y Y Y P[ Y x v]( p) P[ Y x v] p P[ Y x v] P[ N x v] F ( x v) P[ Y x v] P[ N x v] F ( x v) F F ( x v)( p) F ( x v) p Y N N dfy Y N ( x v)( p) N ( x v) p dx x ( xv) ( xv) N e Y e ( p) e p N N THE EPECTED VALUE expeced vlue or me o rdom vrible : E[ ] ( ) d exiss i E[ ] ( ) d Exmple: Me o Uiorm Rdom Vrible b E[ ] d b b b b I he pd is symmeric bou poi m: ( m x) ( m x) he E [] = m. Exmple: Me o Expoeil Rdom Vrible ( ) x [ ] x x x x e x E x e dx xe e dx Expeced Vlue o Y = g () : E[ Y ] g dx Exmple: Expeced Vlues o Siusoid wih Rdom Phse Y cos( ) where, d re coss; is uiorm i [, ]. expeced vlue o Y d expeced vlue o he power o Y, Y :
E[ Y ] E[ cos( )] cos( ) d si( ) E Y E E [ ] [ cos ( )] cos( ) Some properies: cos( ) d (i) E[] c c where c is cos. (ii) E[ c ] ce[ ] (iii) E gk( ) E[ gk( )] k k Vrice o VAR[ ] E ( m ) E[ ] m m E[ ] Sdrd Deviio: STD[ ] VAR[ ] Exmple: Vrice o uiorm RV b b ( b ) m E ( m) x dx b b Exmple: Vrice o he Gussi RV ( xm) ( xm) dx e dx e dx Diereie wih respec o : ( xm) ( xm) ( x m) ( ) 3 e dx x m e dx The h mome o he rdom vrible : E[ ] x dx
IMPORTANT RVs The uiorm RV Pd: x b b x, x b Cd x x F x b b x b me d vrice: The expoeil RV ( ) [ ] b VAR[ ] b E Pd: Cd: x x e x x F x e x me d vrice: E[ ] VAR[ ] memoryless propery: P[ h ] P[ h] proo: ( h) P[{ h} { }] P[ h] e h P[ h ] e P[ h] P[ ] P[ ] e The Gussi (Norml) Rdom Vrible Pd: ( xm) e x
Cd: ( um) F P[ x] e du x ( xm)/ x m u m F e d where x x e d pd o Gussi RV wih m = d =. Exmple: Show h Gussi pd iegres o oe. x x y ( x y ) e dx e dx e dy e dxdy x r cos, y rsi x y r dxdy rdrd ( x y ) r r e dxdy e rdrd e rdr The Q-ucio: Q e d x The Gmm Rdom Vrible x e Pd: For, x ( ) where ( ) z x z x e dx z he Gmm ucio properies: ( z ) z( z) z ( m ) m! m oegive ieger. Whe m, ieger : m-erlg RV
x y dx x e dx y x y e dy ( ) ( ) dx The Cuchy Rdom Vrible / hs o momes becuse he iegrls do o coverge. x FUNCTIONS OF A RANDOM VARIABLE Give Y = g() problem is o id he pd o Y i erms o he pd o. Exmple: Y b y b y b FY( y) P[ Y y] P[ b y] P[ ] F i y b y b P[ ] F i y b Y( y) y b Y( y) y b Y( y)
Exmple: Y = y P[ Y y] P y y y y FY ( y) F( y ) F( y ) y dy df y d y df y d y Y ( y) d y d y dy y y y y y I is Gussi wih me m =, d SD =, x y y Y e y e ( y) e y y (chi-squre rv wih oe degree o reedom) Geerl Cse Soluios o y g : x, x,..., x Le = 3,
P[ y Y y dy] P[ x x dx ] P[ x dx x ] P[ x x dx ] 3 3 3 ( y) dy ( x ) dx ( x ) dx ( x ) dx Y 3 3 ( x) ( x3) Y ( y) dy / dx dy / dx dy / dx xx xx xx3 I geerl, ( y) Y k ( xk) g( x ) k Previous exmple: Y = y x soluios or y : x y x y g x g( x ) y g( x ) y g( x ) g( x ) y y y y ( x) e e e Y ( y) g( x) g y y y Exmple: Y = cos() : uiormly disribued i [,], x y cos Two soluios i [, ] x cos ( y) x cos ( y) g si g( x) si cos ( y) g si cos ( y) cos ( y) y cos( ) si( ) cos ( ) y g( x ) g( x ) y
Y ( y) y y y y THE MARKOV AND CHEBYSHEV INEQUALITIES Mrkov iequliy: E [ ] P[ ] is o-egive. This boud is useul i we do hve y iormio bou he RV excep is me vlue. Proo: E[ ] ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d P[ ] I oly he me d vrice re kow: The Chebyshev iequliy: igher boud h Mrkov. P m Proo: Le ED [ ] D ( m) Mrkov iequliy E[ D ] Exmple: Muli-user compuer sysem; me respose ime m = 5 s sdrd deviio = 3 s Esime he probbiliy h he respose ime is more h 5 secods rom he me. Chebyshev iequliy 9 5 s P 5 5.36 5 The Chero boud: Le A = { }. Idicor ucio o A, IA() P[ ] I ( ) ( ) d A
Choose he boud or he idicor ucio s I A( ) or ll E[ ] I A( ) ( ) d ( ) d Le he boud be : Mrkov iequliy. I e s s( ) A( ) s( ) s s s s I A( ) ( ) d e ( ) d e e ( ) d e E[ e ] s s P[ ] e E[ e ] : he Chero boud The Chrcerisic Fucio: Coiuous RVs j jx ( ) E[ e ] e dx Fourier rsorm o he pd jx ( ) e d Iverse Fourier rsorm Discree RVs: j xk ( ) p ( x ) e k k ( ) p ( k) e jk : ieger-vlued RVs k jk p( k) ( ) e d Exmple: Expoeil RV x jx ( j) x ( ) e e dx e dx j Exmple: Geomeric RV k jk j k p ( ) pq e p ( qe ) qe k k j
The mome Theorem: d E [ ] ( ) j d Proo: Tylor series expsio o j x e jx ( jx) e jx...! jx ( jx) ( ) e dx jx... dx! ( j) j x dx x dx...! ( j) ( j) ( ) je[ ] E[ ]... E[ ]...!! Compre wih he Tylor series expsio o ( ) bou = : ( ) () ( j) ( ) d E[ ] E[ ] () ( )!! j j d Exmple: me d vrice o expoeilly disribued rdom vrible j ( ) ( ) E [ ] () j ( j) j ( ) E[ ] () VAR[ ] 3 ( j) j RELIABILITY The relibiliy ime is deied s he probbiliy h he compoe, subsysem, or sysem is sill ucioig ime : R( ) P[ T ] T: lieime o sysem R( ) P[ T ] FT ( ) FT () : cd o T. R( ) ( ) T The me ime o ilure (MTTF) is give by he expeced vlue o T: E[ T ] ( ) d R( ) d R( ) d T The codiiol cd o T give h T > :
FT ( x T ) P[ T x T ] FT FT( ) FT ( ) x x T T ( x T ) x F ( ) T R() The ilure re ucio r() : r( ) T ( T ) R () r() d = he probbiliy h compoe h hs ucioed up o ime will il i he ex d secods. Exmple: Expoeil Filure Lw cos ilure re ucio : R() r( ) iiil codiio R() R () dr R() d l R( ) e R R() Relibiliy o Sysems ( ) e : T is expoeilly disribued RV. T A s : he eve sysem ucioig ime A j : he eve jh compoe is ucioig ime Series Coecio: P[ A ] P[ A A... A ] P[ A ] P[ A ]... P[ A ] s R( ) R ( ) R ( )... R ( ) I sysems hve cos ilure res: ( ) i R e R( ) e i (... ) Prllel Coecio: The sysem will o be ucioig i d oly i ll he compoes hve iled.
P[ A ] P[ A ] P[ A ]... P[ A ] c c c c s R( ) R ( ) R ( )... R ( ) R( ) R ( ) R ( )... R ( ) Exmple: Compre he relibiliy o sigle-ui sysem gis h o sysem h operes wo uis i prllel. Assume ll uis hve expoeilly disribued lieimes wih re. sigle-ui sysem: R () s e wo uis i prllel: R ( ) ( e )( e ) e ( e ) e p R ( ) R ( ) p s