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Global+Optimization+Algorithms+Theory+and+Application_Part25

Global+Optimization+Algorithms+Theory+and+Application_Part25...

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28.4 Some Discrete Distributions 481 distribution are listed in Table 28.3 43 and examples for its PDF and CDF are illustrated in Figure 28.3 and Figure 28.4 . parameter definition parameters λ = μt> 0 (28.120) PMF P ( X = x ) = f X ( x ) = ( μt ) x x ! e μt = λ x x ! e λ (28.121) CDF P ( X x ) = F X ( x ) = Γ ( k +1 ) k ! = x i =0 e λ λ i i !! (28.122) mean EX = μt = λ (28.123) median med ≈⌊ λ + 1 3 1 5 λ (28.124) mode mode = λ (28.125) variance D 2 X = μt = λ (28.126) skewness γ 1 = λ 1 2 (28.127) kurtosis γ 2 = 1 λ (28.128) entropy H( X ) = λ (1 ln λ ) + e λ k =0 λ k ln( k !) k ! (28.129) mgf M X ( t ) = e λ ( e t 1) (28.130) char. func. ϕ X ( t ) = e λ ( e i t 1) (28.131) Table 28.3: Parameters of the Poisson distribution. 4 8 12 16 20 24 28 l =1 l =6 l =20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 f (x) X x l =1 l =3 l =6 l =10 l =20 l Figure 28.3: The PMFs of some Poisson distributions Poisson Process The Poisson process 44 [1914] is a process that obeys the Poisson distribution – just like the example of the telephone switchboard mentioned before. Here, λ is expressed as the product of the intensity μ and the time t . μ normally describes a frequency, for example μ = 1 min . Both, the expected value as well as the variance of the Poisson process are λ = μt . In 43 The Γ in Equation 28.122 denotes the (upper) incomplete gamma function. More information on the gamma function Γ can be found in Section 28.10.1 on page 532 . 44 http://en.wikipedia.org/wiki/Poisson_process [accessed 2007-07-03]
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482 28 Stochastic Theory and Statistics 0 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x F (x) X 5 10 15 20 25 30 l =1 l =3 l =6 l =10 l =20 discontinuous point 0.2 Figure 28.4: The CDFs of some Poisson distributions Equation 28.132 , the probability that k events occur in a Poisson process in a time interval of the length t is defined. P ( X t = k ) = ( μt ) k k ! e μt = λ k k ! e λ (28.132) The probability that in a time interval [ t,t + Δt ] 1. no events occur is 1 λΔt + o ( Δt ). 2. exactly one event occurs is λΔt + o ( Δt ). 3. multiple events occur o ( Δt ). Here we use an infinitesimal version the small- o notation. 45 The statement that f o ( ξ ) ⇒| f ( x ) |≪| ξ ( x ) | is normally only valid for x →∞ . In the infinitesimal variant, it holds for x 0. Thus, we can state that o ( Δt ) is much smaller than Δt . In principle, the above equations imply that in an infinite small time span either no or one event occurs, i.e., events do not arrive simultaneously: lim t 0 P ( X t > 1) = 0 (28.133) The Relation between the Poisson Process and the Exponential Distribution It is important to know that the (time) distance between two events of the Poisson process is exponentially distributed (see Section 28.5.3 on page 489 ). The expected value of the number of events to arrive per time unit in a Poisson process is EX pois , then the expected value of the time between two events 1 EX pois . Since this is the excepted value EX exp = 1 EX pois of the exponential distribution, its λ exp -value is λ exp = 1 EX exp = 1 1 EX pois = EX pois . Therefore, the λ exp -value of the exponential distribution equals the λ pois -value of the Poisson distribution λ exp = λ pois = EX pois . In other words, the time interval between (neighboring) events of the 45 See Section 30.1.3 on page 550 and Definition 30.16 on page 551
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