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Exhibit 8.7 Poisson probability distribution with λ = 5
4 or 5 are equal, approximately 0.175 each. This must be taken in the context of
an average arrival rate of
5. It makes sense that if the average arrival rate
is 5, the values near 5 will have a higher probability than those that are distant,
for example 11, which has a probability of less than 0.01. Thus, the further away
an outcome is from the average arrival rate, the smaller the probability of its
occurrence for the Poisson.
5. Similarity in distributions —Many distributions often have another distribution
for which they possess some similarity. Other distributions often represent a
family of distributions. The Beta distribution family is one such Continuous dis-
tribution. In fact, the Uniform is a member of the Beta family. The Poisson is
closely related to the Binomial distribution. Like the Poisson, the Binomial is
also a Discrete distribution and provides the probability of a speciﬁc number of
successes, k ,in n trials of an experiment that has only two outcomes, and where
the probability of success for an individual trial (experiment) is p . Thus, we could
ask the probability question—what is the probability that I will get 9 heads ( k )in
20 tosses ( n ) of a coin, where the probability of a head is 0.5 ( p ). In Exhibit 8.8
we see that the probability is approximately 0.12. In situations where the results
of a trial can only be one of two outcomes, the Binomial is a very useful discrete
distribution.
6. Other important characteristics of distributions
λ =
a. Shape . Most of the distributions we have discussed, thus far, are unimodal ,
that is, they possess a single maximum value. Only the Uniform is not
unimodal. It is also possible to have bimodal, trimodal, and multimodal
distributions, where the modes need not be equal, but are merely localized
maximum values. For example, in Exhibit 8.9 we see a bimodal distribution
where one local maximum is the outcome 4 and the other outcome is 12. This
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