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ISEN 609 Lecture 8

# ISEN 609 Lecture 8 - 8 Poisson Process first some...

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8. Poisson Process: first some background Poisson distribution: Where does this formula come from? Poisson distribution is an “approximation” to the Binomial distribution. Y Bin [ n, p ] = P ( Y = k ) = ± n k ² p k (1 - p ) n - k , k = 0 , 1 , . . . , n X Pois [ λ ] = P ( X = k ) = e - λ λ k k ! , k = 0 , 1 , . . .

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Poisson Approximation to Binomial Let’s look at what happens to Bin [ n , p ] as n gets large and p get small (such that the product np is held constant - call it ): λ n → ∞ : n ( n - 1) · · · ( n - k + 1) n k 1 , ± 1 - λ n ² n e - λ , ± 1 - λ n ² k 0 P ( Y = k ) λ k e - λ k ! , k = 0 , 1 , . . . P ( Y = k ) = ± n k ² p k (1 - p ) n - k = n ! k !( n - k )! ³ λ n ´ k ³ 1 - λ n ´ n - k = n ( n - 1) · · · ( n - k + 1) n k λ k k ! (1 - λ n ) n (1 - λ n ) k
“Memoryless” property We say that a random variable has the memoryless property if: What does this property say? Suppose the random variable Z represents the lifetime of an engineered product P ( Z > t + s | Z > s )= P ( Z > t )

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Exponential Distribution Suppose . Then for In fact, the exponential is the only continuous r.v. with this property. P ( Z > t + s | Z > s ) = P ( Z > t + s, Z > s ) P ( Z > s ) = P ( Z > t + s ) P ( Z > s ) = e - μ ( t + s ) e - μs = e - μt = P ( Z > t ) Z exp ( μ ) t, s > 0
Exponential Distribution Let X and Y be independent random variables with parameters , respectively. Find the distribution of min( X,Y ) max( X,Y ) X + Y Find P(X > Y) μ and ν

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Counting Processes <--> Point Process Many of the models we develop involving characterizing occurrences that happen over time. Suppose we are counting the number of occurrences of a particular event over time, starting at a point t = 0. Let N ( t ) be the number of events we have observed up to time t ; i.e., the number of events in [0, t ]. is called a counting process . Counting processes make up some of the most common models in engineering - arrivals to a service system, flaws in a manufacturing process, bugs in a piece of software, etc. { N ( t ) , t 0 }
And Now: Introducing The Poisson Process If random occurrences happen over time according to certain assumptions - namely, stationarity, independent increments, and orderliness - and if we start counting at time 0, and if 0 < P ( N ( t ) = 0) < 1, then there exists a positive number such that Note that and therefore and λ P ( N ( t ) = n ) = ( λ t ) n e - λ t n ! , n = 0 , 1 , 2 , . . . , t > 0 N ( t ) Pois [ λ t ] , E [ N ( t )] = λ t λ = E [ N (1)]

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Relationship between Counts, Interarrival Times, and Occurrence Times Note that if we know the counting process , then we can determine the sequence of interarrival times as well as the sequence of occurrence times (Ross calls these waiting times) ; these are related by = n th jump in the counting process So to describe a point process model, we need to make assumptions about a counts, or waiting times, or interarrival times. { N ( t ) , t 0 } { S i ,i =1 , 2 , . . . } { T i , 2 , . . . } S n S n = n ± i =1 T i
The Poisson Process A orderly , stationary counting process with independent increments is called a Poisson process.

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ISEN 609 Lecture 8 - 8 Poisson Process first some...

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