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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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