Fault Tolerant Computing. Fault Tolerant Computing CS 530 Probabilistic methods: overview

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1 Probably 1/19/ CS 53 Probablsc mehods: overvew Yashwa K. Malaya Colorado Sae Uversy 1 Probablsc Mehods: Overvew Cocree umbers presece of uceray Probably Dsjo eves Sascal depedece Radom varables ad dsrbuos Dscree dsrbuos: Bomal, Posso Couous dsrbuos: Gaussa, Expoeal, Webull Sochasc processes Markov Posso YKM 1

2 Probably 1/19/ Bascs Probably of a eve A (bewee ad 1) P { A} N f A occurs mes amog N equally lkely oucomes. Ex: Roll of a de 3 odd}.5 If more formao s avalable, probably of he same eve chages. If we kow de s loaded, perhaps odd}. 3 Bascs () Uo: A A} + A Ex: Roll of a de oucome eve oucome 3} If A ad B are dsjo, I.e. f AB φ, AU A} + A}1-A} A AB eve} + 3} eve 3} + B 4 YKM

3 Probably 1/19/ Codoal probably Bascs (3) A A for > If A ad B are depede, A A}. The A A}. If A ca be dvded o dsjo A, 1,..,, he B A } A }. 5 Radom Varables X: radom varable (ex: hegh of a radomly chose sude) x: oe specfc value (say 5 9 ) f ( x) dx F( x) E( X ) couous x X x + dx} x x m x max x m f ( x) dx x f ( x) dx dscree max m max p( x ) m p( x ) x p( x ) YKM 3

4 Probably 1/19/ Dsrbuos, Bomal Noe ha x max xm Major dsrbuos: Dscree: Boomal, Posso Couous: Gaussa, expomeal Bomal (success/falure) max f ( x) dx 1 p( x ) 1 m Prob. of r occurreces rals, prob. of oe success beg p r r f ( r) p (1 p) for r,, r cdeally! Cr r r!( r)! 7 Dsrbuos: Posso Posso: also dscree. λoccurrece rae of somehg. Probably of r occurreces me s r ( λ ) e f ( r) r! [ also somemes f ( x) x λ e x! ] 8 YKM 4

5 Probably 1/19/ Dsrbuos: Gaussa 189 Couous. Also ermed Normal (Laplaca Frace 1774 ) ( x µ ) σ 1 f ( x) e, πσ.8.7 Bell-shaped curve µ 7 σ 5 x + σ :sadard devao Desy µ 7 σ 1 ( varace) µ : mea Grades 9 Normal dsrbuo () Tables for ormal dsrbuo are avalable, ofe erms of sadardzed varable z(x- µ)/σ. (µ-σ, µ+σ) cludes 8.3% of he area uder he curve. (µ-3σ, µ+3σ) cludes 99.7% of he area uder he curve. Ceral Lm Theorem: Sum of a large umber of depede radom varables eds o have a ormal dsrbuo. 1 YKM 5

6 Probably 1/19/ Expoeal & Webull Ds. Expoeal: Couous. λ: ex or falure rae. Pr{ex he good sae durg (, +d)} e - λ λ d Desy fuco f() λ e - λ < Prob. of o ex (,) s 1-F(). (Relably) Webull Dsrbuo: -parameer geeralzao of expoeal. Beer f may cases. More laer. f() λ. 3 7 λ Sae e -λ λ 5 1/ λ 1 15 me 11 Varace & Covarace Varace: a measure of spread Var{X} E[X-µ x ] Sadard devao (Var{x}) 1/ σ sadard devao (usually for ormal ds) Covarace: a measure of sascal depedece Cov{X,Y} E[(X-µ x )(Y-µ y )] Correlao coeffce: ormalzed ρ xy Cov{X,Y}/ σ x σ y Noe ha < ρ xy <1 1 YKM

7 Probably 1/19/ Sochasc Processes Sochasc process: ha akes radom values a dffere mes. Ca be couous me or dscree me Markov process: dscree-sae, couous me. Traso probably from sae o sae j depeds oly o sae (memory-less) Markov cha: dscree-sae, dscree me Posso process: A Markov coug process N(),, N() s he umber of arrvals up o me. 13 Posso Process: defo Posso process: A Markov coug process N(),, N() s he umber of arrvals up o me. N() a arrval me }λ No smulaeous arrvals λ λ λ λ 1 arrvals 14 YKM 7

8 Probably 1/19/ Posso process: aalyss λ λ λ λ 1 P process arrvals sae } P ( + ) P [1 λ ] P ( + ) P P dp P d Soluo : l( P ) + C P C e P e Sce P () 1, C 1, 15 Posso Process: more Soluo : l( P ) + C P C e P e Sce P () 1, C 1, I geeral, dp P + λp d Solvg recursvely, ( λ) P! e 1,1,.. 1 YKM 8

9 Probably 1/19/ Posso Process: Tme bewee Two Eves T1 T Frs arrval me T > } o arrval (, )} e Thus F( ) T } 1 e f λ λe Expoeal dsrbuo 17 YKM 9