GE 331-Lecture 13

GE 331-Lecture 13 - Normal approximation to Binomial and...

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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 1 Normal approximation to Binomial and Poisson • Recall that a binomial r. v. X has mean and variance: m= np, s 2 =np(1-p) • Standardization: when np>5, n(1-p)>5 • Recall a Poisson r.v. X has mean and variance μ = = λ • Standardization : when λ >5 ) 1 ( p np np X X Z = = σ μ λ = = X X Z
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 2 Exponential distribution • Poisson process : counts the # of events occur randomly (# of calls to a service center), λ = average number of events / unit time • Interested in the waiting time for the next event waiting time
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 3 Exponential distribution (cont) X : the waiting time until 1 st call comes • What is the probability distribution of X? e.g., P(X > x) for some x 0 ? X > x no call in time interval [ 0, x ] Let N x be the number of calls arriving in [ 0, x ], then P(X > x)=P(N x =0)
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 4 Exponential distribution (cont) • Recall that for a Poisson process with arrival rate λ , • cdf of X ... 2 , 1 , 0 ! ) ( ) ( = = = n e n x n N P x n x ! 0 ) ( ) 0 ( ) ( 0 x x x e e x N P x X P = = = = > 0 , 1 ) ( 1 ) ( ) ( = > = = x e x X P x X P x F x
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 5 Exponential distribution (cont) The cdf for an exponential distribution with parameter λ is given by The pdf for the exponential distribution Waiting time is non-negative. So for x<0, f(x)=0, F(x)=0 0 1 ) ( = x e x F x λ 0 ) ( ' ) ( = = x e x F x f x
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 6 Exponential distribution (cont) • For an exponential distribution with arrival rate λ Cumulative distribution function Probability density funciton Mean E(X)=1/ λ , variance V(X)=1/ λ 2 , standard deviation =1/ λ < = = 0 0 0 1 ) ( ) ( x x e x X P x F x λ < = = 0 0 0 ) ( ' ) ( x x e x F x f x 0 , ) ( = > x e x X P x
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 7 Exponential distribution (cont) • Memoryless property, for any s, t 0 ) ( ) exp( )) ( exp( ) ( ) ( ) ( ) , ( ) | ( t X P e s s t s X P s t X P s X P s X s t X P s X s t X P t > = = + = > + > = > > + > = > + > λ t t s
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IE 300/GE 331 Lecture 13 Negar Kiyavash, UIUC 8 Exponential distribution (cont) Example: Suppose the number of calls arriving at a service
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GE 331-Lecture 13 - Normal approximation to Binomial and...

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