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Unformatted text preview: Extra Please use a cover page to put your name, ID and Section information with your assignment. () January 29, 2011 1 / 25 Poisson Random Variables Suppose we monitor the occurrence of a centain incident in a fixed period of time. earthquakes over 5 in the Richter scale with epicenter in BC in 2011. () January 29, 2011 2 / 25 Poisson Random Variables Suppose we monitor the occurrence of a centain incident in a fixed period of time. earthquakes over 5 in the Richter scale with epicenter in BC in 2011. number of visits to a popular website in the next hour. () January 29, 2011 2 / 25 Poisson Random Variables Suppose we monitor the occurrence of a centain incident in a fixed period of time. earthquakes over 5 in the Richter scale with epicenter in BC in 2011. number of visits to a popular website in the next hour. parties you will be invited in 2011. () January 29, 2011 2 / 25 Poisson Random Variables Suppose we monitor the occurrence of a centain incident in a fixed period of time. earthquakes over 5 in the Richter scale with epicenter in BC in 2011. number of visits to a popular website in the next hour. parties you will be invited in 2011. () January 29, 2011 2 / 25 Poisson Random Variables Suppose we monitor the occurrence of a centain incident in a fixed period of time. earthquakes over 5 in the Richter scale with epicenter in BC in 2011. number of visits to a popular website in the next hour. parties you will be invited in 2011. The variable of interest X can be the number of occurrences of the incident in the given time interval. () January 29, 2011 2 / 25 Poisson Random Variables Under certain qualitative assumptions (listed in the next slide) the pmf of such a random variable X is p ( x ) = P ( X = x ) = λ x x ! exp { λ } , x = 0, 1, 2, . . . The parameter λ > 0 is called rate and represents the average number of occurrences of the incident in the given time interval. () January 29, 2011 3 / 25 Poisson Random Variables The range of a Poisson distributed random variable is always { 0, 1, 2, . . . } (open ended). It has a single parameter λ which is always positive. Since exp ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + · · · = ∞ ∑ x = t x x ! we have ∞ ∑ x = p ( x ) = ∞ ∑ x = λ x x ! exp { λ } = 1. () January 29, 2011 4 / 25 Poisson Random Variables The range of a Poisson distributed random variable is always { 0, 1, 2, . . . } (open ended). It has a single parameter λ which is always positive. Since exp ( t ) = 1 + t + t 2 2 ! + t 3 3 ! + t 4 4 ! + · · · = ∞ ∑ x = t x x ! we have ∞ ∑ x = p ( x ) = ∞ ∑ x = λ x x ! exp { λ } = 1. That is, the p ( x ) given earlier has all properties for a pmf. () January 29, 2011 4 / 25 Assumptions lead to Poisson distribution Denote N ( a , b ) the number of occurrences of the incident of interest in the time interval [a, b)....
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This note was uploaded on 04/07/2011 for the course STAT 302 taught by Professor Dr.chen during the Spring '11 term at UBC.
 Spring '11
 Dr.Chen
 Probability

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