Hazard and Reliabiliy Funcions, Failure Raes ECE 313 Probabiliy wih Engineering Applicaions Lecure 20 Professor Ravi K. Iyer Dep. of Elecrical and Compuer Engineering Universiy of Illinois a Urbana Champaign Iyer - Lecure 20 ECE 313 Fall 2013
Announcemens The eamples shown on he board will no be necessarily posed online, so i is bes if you ake noes in he class. There will be weekly quizzes from he ne week on he eamples shown in he class, homeworks, and in-class projecs. Quiz 1: ne Tuesday, November 12 Topics covered in Lecures 18-19: Covariance and Limi Theorems Inequaliies and CLT Homework 8 Soluions will be posed his Thursday Iyer - Lecure 20 ECE 313 Fall 2013
Today s Topics Review of Join and Condiional Densiy Funcions Hazard Funcion Reliabiliy Funcion Insananeous Failure Rae Eamples Iyer - Lecure 20 ECE 313 Fall 2013
Insananeous Failure Rae or Hazard Rae Hazard measures he condiional probabiliy of a failure given he sysem is currenly working. The failure densiy pdf measures he overall speed of failures The Hazard/Insananeous Failure Rae measures he dynamic insananeous speed of failures. To undersand he hazard funcion we need o review condiional probabiliy and condiional densiy funcions very similar conceps Iyer - Lecure 20 ECE 313 Fall 2013
Review of Join and Condiional Densiy Funcions We define he join densiy funcion for wo coninuous random variables X and Y by: φ, yddy P < X + d, y < Y y + dy The cumulaive disribuion funcion associaed wih his densiy funcion is given by: y Φ, y φ, yd dy P < X, < Y y Iyer - Lecure 20 ECE 313 Fall 2013
Parallelepiped Densiy Funcion A parallelepiped densiy funcion is shown below: b < X a + b / 2 d < Y c + d / 2 The probabiliy ha and is given by: Pb < X a + b 2, d < Y c + d 2 Φ a + b 2, c + d 2 a+b/2 c+d /2 a d 1 a bc d dyd 1 a bc d c + d 2 da + b 2 b 1 4 Iyer - Lecure 20 ECE 313 Fall 2013
P AB Review of Join and Condiional Densiy Funcions Con d P A P B Now if we associae random variable X wih A and random variable Y wih B hen, P < P y < P < y + d y + dy + d, y Thus join probabiliy f d g y dy < y y + dy φ, y d dy φ, y f g y Remember random variables X and Y are independen if heir join densiy funcion is he produc of he wo marginal densiy funcions. Iyer - Lecure 20 ECE 313 Fall 2013
Review of Join and Condiional Densiy Funcions Con d If evens A and B are no independen, we mus deal wih dependen or condiional probabiliies. Recall he following relaions We can epress condiional probabiliy in erms of he random variables X and Y PAB PAPB A PB A PAB PA Py < Y y + dy < X + d P < X + d, y < Y y + dy P < X + d The lef-hand side defines he condiional densiy funcion for y given, which is wrien as h y φ, y f Iyer - Lecure 20 ECE 313 Fall 2013
Review of Join and Condiional Densiy Funcions Con d Similarly, he condiional densiy funcion for given y is w φ, y y g y Now we use his o deermine hazard funcion as a condiional densiy funcion Iyer - Lecure 20 ECE 313 Fall 2013
Hazard Funcion: From Condiional Densiy Definiion The ime o failure of a componen is he random variable T. Therefore he failure densiy funcion is defined by P < T + d f d Someimes i is more convenien o deal wih he probabiliy of failure beween ime and +d, given ha here were no failures up o ime. The probabiliy epression becomes P < T + d T > P < T + d PT > a where P T > 1 PT < 1 F Iyer - Lecure 20 ECE 313 Fall 2013
Hazard Funcion The condiional probabiliy on he lef side a gives rise o he condiional probabiliy funcion z defined by z d 0 The condiional funcion is generally called he hazard. Combining a and b: P lim < T + d The main reason for defining he z funcion is ha i is ofen more convenien o work wih han f. d f z 1 F T > b Iyer - Lecure 20 ECE 313 Fall 2013
Iyer - Lecure 20 ECE 313 Fall 2013 Hazard Funcion For eample, suppose ha f is an eponenial disribuion, he mos common failure densiy one deals wih in reliabiliy work. Thus, an eponenial failure densiy corresponds o a consan hazard funcion. Wha are he implicaions of his resul? λ λ λ λ λ λ λ λ e e F f z e F e F e f 1 1 1
The Reliabiliy Funcion Le he random variable X be he lifeime or he ime o failure of a componen. The probabiliy ha he componen survives unil some ime is called he reliabiliy R of he componen: where F is he disribuion funcion of he componen lifeime, X. The componen is assumed o be working properly a ime 0 and no componen can work forever wihou failure: i.e. R P X > 1 F R 0 1 and lim R 0 R is a monoone non-increasing funcion of. For less han zero, reliabiliy has no meaning, bu: someimes we le R1 for <0. F will ofen be called he unreliabiliy. Iyer - Lecure 20 ECE 313 Fall 2013
The Reliabiliy Funcion Con d Consider a fied number of idenical componens, N 0, under es. Afer ime, N f componens have failed and N s componens have survived N f + N s N 0 The esimaed probabiliy of survival: Pˆ survival Ns N 0 Iyer - Lecure 20 ECE 313 Fall 2013
Iyer - Lecure 20 ECE 313 Fall 2013 The Reliabiliy Funcion Con d In he limi as N 0, we epec survival o approach R. As he es progresses, N s ges smaller and R decreases. Pˆ 0 0 0 0 1 N N N N N N N R f f s
The Reliabiliy Funcion Con d N 0 is consan, while he number of failed componens N f increases wih ime. Taking derivaives: R' N f is he rae a which componens fail N 0 As, he righ hand side may be inerpreed as he negaive of he failure densiy funcion, f Δ R' f Noe: is he uncondiional probabiliy ha a componen will fail in he inerval, + Δ 1 N 0 N' f F X Iyer - Lecure 20 ECE 313 Fall 2013
Iyer - Lecure 20 ECE 313 Fall 2013 Insananeous Failure Rae If we know for cerain ha he componen was funcioning up o ime, he condiional probabiliy of is failure in he inerval will in general be differen from This leads o he noion of Insananeous failure rae. Noice ha he condiional probabiliy ha he componen does no survive for an addiional inerval of duraion given ha i has survived unil ime can be wrien as: f Δ R F F X P X P G Y + > + < <
Iyer - Lecure 20 ECE 313 Fall 2013 Insananeous Failure Rae Con d Definiion: The insananeous failure rae h a ime is defined o be: so ha: h represens he condiional probabiliy ha a componen surviving o age will fail in he inerval,+. The eponenial disribuion is characerized by a consan insananeous failure rae: lim 1 lim 0 0 R f h R R R R F F h + + λ λ λ λ e e R f h
Insananeous Failure Rae Con d Inegraing boh sides of he equaion: h d Using he boundary condiion, R01 Hence: R ep h d 0 0 0 or 0 f d R R R0 R' d R dr R ln R 0 h d Iyer - Lecure 20 ECE 313 Fall 2013
Cumulaive Hazard The cumulaive failure rae, H h d, is referred o as he 0 cumulaive hazard. R ep h d gives a useful heoreical represenaion 0 of reliabiliy as a funcion of he failure rae. An alernae represenaion gives he reliabiliy in erms of H cumulaive hazard: R e H λ If he lifeime is eponenially disribued, hen and we obain he eponenial reliabiliy funcion. Iyer - Lecure 20 ECE 313 Fall 2013
f and h f is he uncondiional probabiliy ha he componen will fail in he inerval,+ ] h is he condiional probabiliy ha he componen will fail in he same ime inerval, given ha i has survived unil ime. h is always greaer han or equal o f, because R 1. f is a probabiliy densiy. h is no. [h] is he failure rae [f] is he failure densiy. To furher see he difference, we need he noion of condiional probabiliy densiy. Iyer - Lecure 20 ECE 313 Fall 2013
Failure Rae as a Funcion of Time Iyer - Lecure 20 ECE 313 Fall 2013
Iyer - Lecure 20 ECE 313 Fall 2013 Condiional Probabiliy Densiy Funcion Le denoe he condiional disribuion of he lifeime X given ha he componen has survived pas fied ime. Then Noe ha Then he condiional failure densiy is: V X V X < >. 0,,, 1 F F F X P dy y f V X. G V Y X <.,,, 1 O F f v X
Condiional Probabiliy Densiy Funcion Con d V X v X The condiional densiy saisfies properies f1 and f2 of a probabiliy densiy funcion pdf and hence is a probabiliy densiy while he failure rae h does no saisfy propery f2 since: 0 lim R ep h d 0 1 Iyer - Lecure 20 ECE 313 Fall 2013
Treamen of Failure Daa Par failure daa generally obained from wo sources: he failure imes of various iems in a populaion placed on a life es, or repair repors lising operaing hours of replaced pars in equipmen already in field use. Compue and plo eiher he failure densiy funcion or he insananeous failure rae as a funcion of ime. The daa: a sequence of imes o failure, bu he failure densiy funcion and he hazard inroduced as coninuous variables. Compue a piecewise-coninuous failure densiy funcion and hazard rae from he daa. This is, a specific approach o he very general engineering problem of how o model a problem from cerain qualiaive knowledge abou he sysem suppored by quaniaive daa. Iyer - Lecure 20 ECE 313 Fall 2013
Treamen of Failure Daa con d Define piecewise-coninuous failure densiy and hazard-rae in erms of he daa. Assume ha our daa describe a se of N iems placed in operaion a ime 0. As ime progresses, iems fail, and a any ime he number of survivors in n. The empirical probabiliy densiy funcion defined over he ime inerval i < i + Δ i, is given by he raio of he number of failures occurring in he inerval o he size of he original populaion, divided by he lengh of he ime inerval: f d [ n i n i + Δ Δ i i ]/ N for i < i + Δ i Iyer - Lecure 20 ECE 313 Fall 2013
Treamen of Failure Daa Con d The daa hazard ins. failure rae over he inerval i i i is he raio of he number of failures occurring in he ime inerval o he number of survivors a he beginning of he ime inerval, divided by he lengh of he ime inerval: z d [ n i n i + Δ Δ Observaion: he failure densiy funcion is a measure of he overall speed a which failures are occurring, whereas he hazard rae z d f d is a measure of he insananeous speed of failure. Noe: boh and have he dimensions of inverse ime generally he ime uni is hours. i z d i Δ i i ]/ n < + Δ The choice of and in he above equaions is unspecified and is bes discussed in erms of he eamples ha follow. i for f d i < i + Δ i Iyer - Lecure 20 ECE 313 Fall 2013
Properies of Densiy and Disribuion Funcions Iyer - Lecure 20 ECE 313 Fall 2013
Consrains on f and z Iyer - Lecure 20 ECE 313 Fall 2013
Hazard Rae Eample Iyer - Lecure 20 ECE 313 Fall 2013
Hazard Rae Eample Con d Iyer - Lecure 20 ECE 313 Fall 2013
Hazard Rae Eample Con d Failure Densiy and Hazard Rae for he Given Daa" Iyer - Lecure 20 ECE 313 Fall 2013