GENG5507-STATS-lecture-week12-Revision.pdf - Risk Reliability and Safety GENG5507 Probability and Statistics Lecture Week 12 REVIEW Dr Gopalan Nair

GENG5507-STATS-lecture-week12-Revision.pdf - Risk...

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Risk, Reliability and Safety GENG5507 Probability and Statistics Lecture Week 12 REVIEW Dr Gopalan Nair School of Mathematics and Statistics 1 Revised: 10/18 GN
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1. Probability, Conditional probability & properties If A aary two events then P(AUB)=P(A)+P(B)-P(A∩B); Conditional probability . For P(B)>0 2 ( ) ( ) ( ) P A B P A B P B = Ç If B 1 , B 2 , …, B n are mde and A is any event then, ( ) ( ) ( ) ( ) P A P A B P A B P A B 1 2 n = + + + ! ! " ! If B 1 , B 2 , …, B n is a partition (mde) of S with P(B i )>0 and A is any event then, ( ) ( ) ( ) ( ) ( ) ( ) ( ) P A P A B P B P A B P B P A B P B 1 1 2 2 n n = + + + ! P A B C ( ) = P ( C ) P ( B | C ) P ( A | B C )
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2. Random variable, pdf & cdf f(x): probability density function (pdf) of a continuous random variable X , if f(x) ³ 0 for all x; 3 ( ) f x dx = ¥ ¥ ò 1 ( ) ( ) b a between x f under area dx x f b X a P b a & ) ( = = < < ò The cumulative distribution function, cdf, F(x): ( ) ( ) ò ¥ - = £ = x dy y f x X P x F ) (
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Mean and Variance For a continuous rv X whose pdf is defined by f X (x), its mean is given by Its variance, var(X) or σ X 2 is given by ( ) X X xf x dx μ ¥ = ò 2 2 var( ) ( ) ( ) X X X X x f x dx s μ ¥ = = - ò 4 Var(X)=E(X 2 ) – [E(X)] 2
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Reliability Function R(x) ( ) { } ( ) X x R x P X x f t dt ¥ = ³ = ò ( ) 1 ( ) R x F x = - 5 h ( t ) = f ( t ) 1 F ( t ) = f ( t ) R ( t ) Failure rate(hazard rate) function
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