app6_buses_eqs

app6_buses_eqs - Application Example 6 (Exponential and...

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Application Example 6 (Exponential and Poisson distributions) ARE THE SEQUENCES OF BUS AND EARTHQUAKE ARRIVALS POISSON? The Poisson Process The Poisson process is the simplest random distribution of points on a line. What makes this model simple and convenient to use is the fact that, in a Poisson process, the number and locations of events in non-overlapping (separate) intervals are independent. This condition is often referred to as “lack of memory” of the process, since it implies that the pattern of past events has no relevance to the pattern of future events. For some applications, the use of the Poisson point process can be justified on theoretical grounds, but in most cases one must verify the plausibility of the Poisson assumption by comparing implications of that assumption with actual data. Two verifications are frequently made: 1. If the point process is Poisson, the distribution of the interarrival time, T, is exponential with mean value 1/ λ where λ is the rate parameter of the process; 2. If the point process is Poisson, the number of events in an interval of duration D has Poisson distribution with mean value λ D. Next we illustrate the use of these validation techniques for the process of bus arrivals at a station. The data sets used for this example are synthetic, but they are representative of patterns one may observe in actual samples. Three synthetic data sets have been generated, each containing 100 arrival times in minutes. The data sets are representative of conditions at three different bus stations: 1
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Data Set 1 refers to bus arrivals at Point A in Figure 1, which is close to the dispatch station. Data Set 2 is representative of conditions at Points B, which is far from the dispatch station and Data Set 3 is collected at Point C, in a downtown area. The arrival times are given in the attached tables and are displayed graphically in Figures 2 and 3. Figure 1: Locations of bus data collection station relative to dispatch point 7:03 7:07 7:11 7:15 7:20 7:24 7:25 7:28 7:33 7:43 7:46 7:47 7:49 7:51 7:53 7:58 8:04 8:10 8:16 8:18 8:20 8:25 8:28 8:33 8:35 8:37 8:44 8:50 8:58 9:01 9:09 9:14 9:15 9:18 9:23 9:27 9:29 9:34 9:38 9:40 9:48 9:51 9:55 10:00 10:06 10:10 10:15 10:17 10:24 10:27 10:31 10:38 10:43 10:45 10:51 10:54 10:57 10:59 11:01 11:11 11:14 11:16 11:21 11:28 11:32 11:37 11:45 11:48 11:51 12:01 12:04 12:13 12:16 12:21 12:26 12:29 12:38 12:45 12:51 12:55 13:00 13:06 13:15 13:21 13:24 13:26 13:28 13:31 13:35 13:42 13:47 13:53 13:55 14:02 14:06 14:08 14:11 14:16 14:21 14:28 Table 1: Arrival times at Station A 7:00 7:11 7:15 7:17 7:26 7:29 7:30 7:32 7:34 7:37 7:39 7:41 7:46 7:54 7:57 8:12 8:15 8:24 8:46 8:46 8:50 9:00 9:01 9:10 9:20 9:31 9:33 9:41 9:43 9:49 9:49 10:04 10:04 10:06 10:17 10:17 10:23 10:26 10:31 10:31 10:38 10:44 10:44 10:51 10:52 10:52 10:58 11:02 11:03 11:06 11:11 11:13 11:19 11:19 11:29 11:35 11:39 11:40 11:57 11:57 11:58 11:58 12:04 12:10 12:11 12:16 12:20 12:28 12:28 12:29 12:30 12:35 12:50 13:04 13:12 13:13 13:16 13:18 13:27 13:31
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app6_buses_eqs - Application Example 6 (Exponential and...

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