Please explain both Poisson and exponential distributions and the difference between them. Please include extensive details for lifesaver. Solution The Poisson distribution - Introduction The Poisson distribution is related to the exponential distribution. Suppose a certain event can occur many times within a unit of time. Denote by x the total number of occurrences within a unit of time. Suppose x is unknown (it is a random variable). If the time elapsed between two successive occurrences of the event has an exponential distribution (and it is independent of previous occurrences), then x has a Poisson distribution. The Poisson distribution - Definition The Poisson distribution is characterized as follows: Definition_ Let x be a discrete random variable. Let its support Rx be the set of positive integer numbers (the natural numbers and ): Rx = Z+ For a Poisson process, hits occur at random independent of the past, but with a known long term average rate of hits per unit time. The Poisson distribution would let us find the probability of getting some particular number of hits. Now, instead of looking at the number of hits, we look at the random variable L (for Lifetime), the time you have to wait for the first hit. The probability that the waiting time is more than a given time value is P(L>t)=P(no hits in time t)=0e0!=et (by the Poisson distribution, where =t). P(Lt)=1et (the cumulative distribution function). We can get the density function by taking the derivative of this: f(x)={et0fort0fort<0 Any random variable that has a density function like this is said to be exponentially distributed. The relation between the Poisson distribution and the exponential distribution is summarized by the following proposition: Proposition_ X (the number of occurrences of an event within a unit of time) has a Poisson distribution with parameter if and only if the time elapsed between two successive occurrences of the event has an exponential distribution with parameter and it is independent of previous occurrences..

1. Please explain both Poisson and exponential distributions and the difference between them.
Please include extensive details for lifesaver.
Solution
The Poisson distribution - Introduction
The Poisson distribution is related to the exponential distribution. Suppose a certain event can
occur many times within a unit of time. Denote by x the total number of occurrences within a
unit of time. Suppose x is unknown (it is a random variable). If the time elapsed between two
successive occurrences of the event has an exponential distribution (and it is independent of
previous occurrences), then x has a Poisson distribution.
The Poisson distribution - Definition
The Poisson distribution is characterized as follows:
Definition_ Let x be a discrete random variable. Let its support Rx be the set of positive integer
numbers (the natural numbers and ):
Rx = Z+
For a Poisson process, hits occur at random independent of the past, but with a known long term
average rate of hits per unit time. The Poisson distribution would let us find the probability of
getting some particular number of hits.
Now, instead of looking at the number of hits, we look at the random variable L (for Lifetime),
the time you have to wait for the first hit.
The probability that the waiting time is more than a given time value is
P(L>t)=P(no hits in time t)=0e0!=et (by the Poisson distribution, where =t).
P(Lt)=1et (the cumulative distribution function). We can get the density function by taking the
derivative of this:
f(x)={et0fort0fort<0
Any random variable that has a density function like this is said to be exponentially distributed.
The relation between the Poisson distribution and the exponential distribution is summarized by
the following proposition:

2. Proposition_ X (the number of occurrences of an event within a unit of time) has a Poisson
distribution with parameter if and only if the time elapsed between two successive occurrences
of the event has an exponential distribution with parameter and it is independent of previous
occurrences.