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# lect14 - THE POISSON PROCESS EXPONENTIAL RANDOM VARIABLES...

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THE POISSON PROCESS EXPONENTIAL RANDOM VARIABLES Outline THE POISSON PROCESS EXPONENTIAL RANDOM VARIABLES MOMENTS MINIMUM OF EXPONENTIALS THE MEMORYLESS PROPERTY THE GAMMA DISTRIBUTION 1 / 14 Xinghua Zheng Lect 14: Exponential and Gamma random variables

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THE POISSON PROCESS EXPONENTIAL RANDOM VARIABLES THE POISSON PROCESS Recall: to model the number of occurrence of some events, e.g., bomb hits or earthquakes, in an interval of time or space, in many situations it is reasonable to assume that, for some λ > 0, 1. The probability that exactly 1 event occurs in an interval of length h is λ h when h is small; 2. The probability that 2 or more events occur in an interval of length h is 0 when h is small; 3. The events occur independently over non-overlapping intervals When these assumptions hold, the total number N ( 1 ) of occurrence in the interval [ 0 , 1 ] is Poisson ( λ ) : (i) Take n large, and for i = 1 , 2 , . . . , n , let X i = 1 if exactly 1 event occurs in [( i - 1 ) / n , i / n ) , and = 0 otherwise (ii) By assumption 2, N ( 1 ) X 1 + X 2 + . . . + X n (iii) By assumptions 1 and 3, X 1 + X 2 + . . . + X n B ( n , λ/ n ) (iv) p = λ/ n is small, so by the Poisson approximation to binomial , N ( 1 ) Poisson ( λ ) What’s the distribution of the total number N ( 2 ) of occurrence in the interval [ 0 , 2 ] ? N ( 3 ) in [ 0 , 3 ] ? In general N ( t ) in [ 0 , t ] ? 2 / 14 Xinghua Zheng Lect 14: Exponential and Gamma random variables
THE POISSON PROCESS EXPONENTIAL RANDOM VARIABLES THE POISSON PROCESS The process N ( t ) ( t 0 ) counting the number of occurrence before time t is called a Poisson process (with rate λ ) For any t > 0, N ( t ) Poisson ( λ t ) Q: If X Poisson ( λ 1 ) , Y Poisson ( λ 2 ) , and they are independent, then Z := X + Y . Let X be the number of occurrence in time interval , and Y be the number of occurrence in time interval For k = 1 , 2 , . . . , let W k be the time when the k th event occurs. W k ’s are called the waiting times For k = 1 , 2 , . . . , let T k be the time elapsed between the ( k - 1 ) th event and the k th event. T k ’s are called the interarrival times T 1 = W 1 For k 2, T k = W k - W k - 1 ; W k = T 1 + . . . + T k 3 / 14 Xinghua Zheng

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