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three basic ways we can model uncertain behavior with probability distributions:
theoretical distributions, expert opinion, or empirically determined distributions.
Here are a few important characteristics to consider about distributions:
1. Discrete Distributions —Recall that distributions can be classiﬁed as either
Discrete or Continuous. Discrete distributions permit outcomes for a discrete,
or countable number of values. Thus, there will be gaps in the outcomes. For
example, the outcomes of arrival of patients at an emergency room hospital dur-
ing an hour of operation are discrete; they are a countable number of individuals.
We can ask questions about the probability of a single outcome value, 4 patients
for example, in a Discrete distribution, as well is the probability of a range of
values, between 4 and 8 customers. The Poisson is a very important Discrete
distribution that we will discuss later in one of our examples. It is restricted to
having integer values.
2. Continuous Probability Distributions or Probability Density functions
Continuous distributions permit outcomes that are continuous over some range.
Probability density functions allow us to describe the probability of events occur-
ring in terms of ranges of numerical outcomes. For example, the probability of
an outcome having values from 4.3 to 6.5 is a legitimate question to ask of a
Continuous distribution. But, it is not possible to ﬁnd the probability of a point
value in a Continuous distribution. Thus, we cannot ask—what is the probability
of the outcome 5 in a Continuous distribution? It is undeﬁned.
3. Discrete and Continuous Uniform Distribution —Exhibits 8.5 and 8.6 show
Discrete and Continuous Uniform distributions, respectively. As you can see in
Exhibit 8.5, the probability of the outcome 7 is 0.2. In the case of the Continuous
Uniform in Exhibit 8.6, we have a distribution from the outcome range of values
4 to 8, and the distribution is expressed as a probability density function .This
relates to our discussion in 2, above. The total area under the Continuous distri-
bution curve is equal to 1. Thus, to ﬁnd the probability of a range of values, say
the range 4 to 6, we ﬁnd the proportion of the area under the distribution that
Exhibit 8.5 Discrete uniform distribution example
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