# Discrete

The discrete distribution can be constructed with Crystal Ball's Custom distribution

Discrete equations

The Discrete distribution describes a variable that can take one of several explicit discrete values {xi} and where a probability weight {pi} is assigned to each value. For example, the number of bridges to be built over a motorway extension or the number of times a software module will have to be re-coded after testing. An example of the Discrete distribution is shown below:

The Discrete distribution is one of the six uses of Crystal Ball's Custom distribution. The other five distributions you can construct with the Custom distribution in Crystal Ball are the Discrete Uniform distribution, the General distribution, the Cumulative Ascending distribution, the Histogram distribution or a combination of different distributions.

#### Uses

##### 1. Probability branching

A Discrete distribution is also particularly useful to describe probabilistic branching. For example, a firm estimates that it will sell Normal(120,10) tons of weed killer next year unless a rival firm comes out with a competing product, in which case it estimates it sales will drop to Normal(85,9) tons. It also estimates that there is a 30% chance of the competing product appearing. This could be modeled by:

Sales = Custom(A1:B2) where the cells A1:B2 contain the formulae:

A1: =Normal (120, 10)

A2: =Normal (85, 9)

B1: 70%

B2: 30%

One can use the Discrete distribution this way only if the Custom distribution is "dynamic".

##### 2. Combining expert opinion

A Discrete distribution can also be used to combine two or more conflicting expert opinions as shown in the following spreadsheet:  Combining_opinions. Also, one can use the Discrete distribution this way only if the Custom distribution is "dynamic".

##### 3. Construct a user-defined discrete distribution

You may wish to use some probability distribution in a Crystal Ball model, but it is not among the distributions Crystal Ball offers. If you know the probability mass function of the discrete distribution, you can use the Discrete distribution to create it. For example:

General principle

Logarithmic distribution

Inverse Hypergeometric distribution

It is not necessary to normalize the probability weights {pi}: Crystal Ball will automatically rescale them to sum to one.

The Discrete Uniform (another use of Crystal Ball's Custom distribution) and the Integer Uniform distributions are special cases of the Discrete distribution where all possible values have the same probability of occurrence.

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