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Exhibit 8.6 Continuous uniform distribution example
is implied by the range. In this case, the range 4 to 6 covers 2 units of a total
interval of 4 (8-4). The area between each successive integer value (4 to 5, 5 to
6, etc.) represents 25% of the area of the entire distribution. Thus, we have 50%
of the area under the curve ([6–4] 0.25
0.5). The density is calculated as the
inverse of the difference between the low and high values of the outcome range
(1/[high range value – low range value]), in our case 0.25 (1/[8–4]). Although the
Uniform describes many phenomena, there is one particular use of the Uniform
that is very interesting and useful. It is often the case that a decision maker simply
has no idea of the relative frequency of one outcome versus another. In this case,
the decision maker may say the following—I just don’t know how outcomes will
behave relative to each other. This is when we resort to the Uniform to deal with
our total lack of specific knowledge and attempt to be fair about our statement
of relative frequency. For example, consider a family reunion to which I invite
100 relatives. I receive regrets (definitely will not attend) from 25, but I have no
idea about the attendance of the 75 remaining relatives. A Discrete Uniform is
a good choice for modeling the 0 to 75 relatives that might attend; any number
of attendees from 0 to 75 is equally likely since we do not have evidence to the
contrary.
4. Specification of a distribution —Distributions are specified by one or more
parameters; that is, we provide a parameter or set of parameters to describe the
specific form the distribution takes on. Some distributions, like the Normal, are
described by a location parameter, the mean (Greek letter
=
μ
), and a dispersion
parameter, the standard deviation (Greek letter
). In the case of the attendees to
our family reunion, the Uniform distribution is specified by a lower and upper
value for the range, 0 to 75. The Poisson distribution has a single parameter, the
average arrival rate , and the rate is denoted by the Greek letter
σ
λ
. Exhibit 8.7
shows a Poisson distribution with a
5. Note that the probability of obtaining
an outcome of 7 arrivals is slightly greater than 0.1, and the probabilities of either
λ =
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