# The Poisson Distribution

Let \$latex alpha\$ be a positive constant. Consider the following probability distribution:

\$latex displaystyle (1) P(X=j)=frac{e^{-alpha} alpha^j}{j!} j=0,1,2,cdots\$

The above distribution is said to be a Poisson distribution with parameter \$latex alpha\$. The Poisson distribution is usually used to model the random number of events occurring in a fixed time interval. As will be shown below, \$latex E(X)=alpha\$. Thus the parameter \$latex alpha\$ is the rate of occurrence of the random events; it indicates on average how many events occur per unit of time. Examples of random events that may be modeled by the Poisson distribution include the number of alpha particles emitted by a radioactive substance counted in a prescribed area during a fixed period of time, the number of auto accidents in a fixed period of time or the number of losses arising from a group of insureds during a policy period.

Each of the above examples canâ€¦

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