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Poisson processes with examples

# Poisson processes with examples - 12.1 Poisson Processes...

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12.1 Poisson Processes For a specified event that occurs randomly in continuous time, an important application of probability theory is in modeling the number of times such an event occurs. The following are several examples of such random phenomenon. Example 1: The number of patients that arrive at a hospital emergency room. Example 2: The number of customers that enter a particular bank. Example 3: The number of accidents at an intersection. Example 4: The number of alpha particles emitted by a radioactive substance. Consider an event that occurs randomly and homogenously in continuous time at an average rate of λ per unit of time. We will refer to the occurrence of the event as a success. If we begin counting successes at time 0, and, for each time, t ≥0 , we let N(t) = number of successes by time t. Properties of N(t): N(0) = 0 N(t)-N(s) is the number of successes in the time interval (s, t] To model the counting process, N(t), we use Bernouilli Trials. A counting process {N(t): t ≥0} is said to be a Poisson process with rate λ if the following 3 conditions hold: Definition Poisson Process a) N(0) = 0 Why does this make sense?

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Poisson processes with examples - 12.1 Poisson Processes...

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