343 / 762
Chapter 8: Discrete Probability Distributions
343
If we let the number of outcomes occurring during a given time interval (or
given region) be denoted by the random variable
X
with the following
assumptions:
1.
The number of outcomes in any given time interval (or region) is
independent of the number of outcomes in any other disjoint time
interval (or region).
2.
The probability that a single outcome will occur during a very short
time interval (or very small region) is proportional to the length of the
time interval (or the size of the region).
3.
The probability does not depend on the number of outcomes occurring
outside the time interval (or region).
4.
The probability that more than one outcome will occur in such a small
time interval (or region) is approximately zero.
Then an experiment satisfying these four conditions is called a Poisson
experiment and we say that
X
(as previously defined) is called a Poisson
random variable.
The function which generates the probability for a Poisson random variable
X
, representing the number of outcomes occurring in a given time interval
(or given region), denoted by
, is given next. Assume that the average
number of outcomes per unit of time (or region) is denoted by
(lambda).