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?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 MARTIN
 Probability, Probability theory, Exponential distribution, Wn, Poisson process

Click to edit the document details