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In this lecture, we introduce
the Poisson process, which is

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a continuous time analog of
the Bernoulli process.

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One way of thinking about
it is as follows.

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Time is continuous, but
conceptually we divide it into

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a very large number of slots.

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And during each slot,
we have a tiny

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probability of an arrival.

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This probability is proportional
to the

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length of the slot.

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Furthermore, we have an
independence assumption for

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the different slots.

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The Poisson process is a very
elegant model of arrival

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processes in continuous time.

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It models many real-world
phenomena.

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And it also has a very clean
mathematical structure that

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allows us to calculate
practically

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every quantity of interest.

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Our development will parallel
our analysis of

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the Bernoulli process.

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For example, we will find the
PMF of the number of arrivals

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during a time interval
and the PDF of the

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time of the kth arrival.

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We will discuss the
memorylessness properties of

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the Poisson process.

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Similar to the case of the
Bernoulli process, this is

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just a consequence of the
independence assumptions that

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we are making.

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We will then exploit these
independence properties to

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argue that the interarrival
times are independent

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exponential random variables.

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And we will conclude with
a comprehensive example.