Changes

A more detailed description of the probability distributions and the required parameters to define them follows.

A more detailed description of the probability distributions and the required parameters to define them follows.

'''Distribution Types'''

==== '''Distribution Types''' ====

 

One of the most familiar continuous probability distributions is the normal distribution as shown in Figure 2.  The normal distribution is also known as the Gaussian distribution or the bell curve. It is a very common distribution in nature and statistics and is based on the central tendency theory.  This is a situation where most items in a group tend towards the average and the further away from the average an event is the less likely it is to occur.  

One of the most familiar continuous probability distributions is the normal distribution as shown in Figure 2.  The normal distribution is also known as the Gaussian distribution or the bell curve. It is a very common distribution in nature and statistics and is based on the central tendency theory.  This is a situation where most items in a group tend towards the average and the further away from the average an event is the less likely it is to occur.  

[[File:Figure 3.svg|center|thumb|523x523px|Figure 3: Normal Distributions]]

[[File:Figure 3.svg|center|thumb|523x523px|Figure 3: Normal Distributions]]

To define a normal distribution only two parameters are needed: the mean and the variance (standard deviation).

To define a normal distribution only two parameters are needed: the mean and the variance (standard deviation).

* '''Flat (Uniform)'''  

===== '''Flat (Uniform)''' =====

 

In a flat distribution any value between a defined minimum and maximum are possible and equally likely.  The outcome within the defined range is completely random.

In a flat distribution any value between a defined minimum and maximum are possible and equally likely.  The outcome within the defined range is completely random.

Figure 4: Flat (continuous uniform) Distribution

[[File:Fig 4.png|center|thumb|517x517px|Figure 4: Flat (continuous uniform) Distribution]]

As can be seen Figure 4, to define this distribution only 2 parameters are required.  They are '''a''' (the minimum possible outcome) and '''b''' (the maximum possible outcome).

As can be seen Figure 4, to define this distribution only 2 parameters are required.  They are '''a''' (the minimum possible outcome) and '''b''' (the maximum possible outcome).

* '''Triangular'''

===== '''Triangular''' =====

The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on knowledge of the minimum and maximum and an "inspired guess" as to the modal value.

The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on knowledge of the minimum and maximum and an "inspired guess" as to the modal value.

===== '''Defined''' =====

 

 

Because of its application as a default distribution to use when little data is available, the triangle distribution is very useful when a model parameter is selected based on solicitation of expert opinion.  The triangular distribution looks something like that is shown in Figure 5.  The parameters required for this distribution are the lowest possible value (the minimum – '''a'''), the most likely value (the mode – '''c'''), and the highest possible value (the maximum – '''b''').

'''            II.     Discrete Distributions'''

The defined distribution is essentially a discrete distribution where any number of sets of outcomes and their probabilities can be created – as long as the probabilities of all outcomes sum to 1.  This can be useful in situations where there is a variable or outcome associated with a worse case scenario (pessimistic), a most likely scenario (realistic), and a best-case scenario (optimistic) as can be seen in Figure 6.  

* '''Defined'''

[[File:Fig 6.png|center|frame|Figure 6 Discrete Defined Distribution]]

The defined distribution is essentially a discrete distribution where any number of sets of outcomes and their probabilities can be created – as long as the probabilities of all outcomes sum to 1.  This can be used in situations where there is a variable or outcome associated with a worse case scenario (pessimistic), a most likely scenario (realistic), and a best-case scenario (optimistic) as can be seen in Figure 6.  

To define this distribution three sets of the outcome and probabilities are required.  These are entered into RRAT by first entering the value followed by a comma and then the probability. Individual outcome and probability sets are separated by semi-colons.

* '''Outcome1,Probability1;Outcome2,Probability2;….OutcomeN,ProbabilityN'''

 

 

 

 

'''Outcome1,Probability1;Outcome2,Probability2;….OutcomeN,ProbabilityN'''

Sets of outcomes are then divided by a semi-colon.  For instance, we may want to make three economic growth forecasts, one based on 1% GDP growth (pessimistic), with a 10% likelihood, one based on 3% GDP growth (realistic) with a likelihood of 70% and one based on 5% GDP growth (optimistic) with a 20% likelihood.  

Sets of outcomes are then divided by a semi-colon.  For instance, we may want to make three economic growth forecasts, one based on 1% GDP growth (pessimistic), with a 10% likelihood, one based on 3% GDP growth (realistic) with a likelihood of 70% and one based on 5% GDP growth (optimistic) with a 20% likelihood.  

These parameters would be entered in RRAT as ''0.01,0.1;0.03,0.7;0.05,0.2''.  Note that the sum of the probabilities of the outcome sets must equal 1 to fully define the distribution.  This function can be used to describe any probabilistic discrete distribution outcome. For instance a coin toss (''heads,0.5;tails,0.5''), a dice role (''1,0.1666666'';2, ''0.1666666;3,0.1666666'';4, ''0.1666666;5,0.1666666;6,0.1666666)'' and many others.

These parameters would be entered in RRAT as ''0.01,0.1;0.03,0.7;0.05,0.2''.  Note that the sum of the probabilities of the outcome sets must equal 1 to fully define the distribution.  This function can be used to describe any probabilistic discrete distribution outcome. For instance a coin toss (''heads,0.5;tails,0.5''), a dice role (''1,0.1666666'';2, ''0.1666666;3,0.1666666'';4, ''0.1666666;5,0.1666666;6,0.1666666)'' and many others.

 

===== '''Fixed''' =====

* '''Fixed'''

Once you have given the variable a description, selected the distribution type, and defined the parameters of the distribution for every item you have completed the first stage. Every time a simulation is run by RRAT a value will be drawn from the probability distributions defined and entered in '''column C''' next to the description. As these are the values that will be used in other formulas in the spreadsheet it is highly recommended that you name these cells appropriately to make them easier to reference.  To determine how to name a cell in Excel, refer to your Excel help function.

Once you have given the variable a description, selected the distribution type, and defined the parameters of the distribution for every item you have completed the first stage. Every time a simulation is run by RRAT a value will be drawn from the probability distributions defined and entered in '''column C''' next to the description. As these are the values that will be used in other formulas in the spreadsheet it is highly recommended that you name these cells appropriately to make them easier to reference.  To determine how to name a cell in Excel, refer to your Excel help function.

 

[[File:Fig 7.png|none|thumb|1160x1160px|Figure 7: Naming a Cell]]

Figure 7: Naming a Cell

 

In the above example (Figure 7) a defined distribution has been used with a low population growth rate of 1% having a 40% probability, medium population growth rate of 3% at 50% probability, and a high population growth rate of 5% at 10% probability.  Each time a simulation is run either 1%, 3%, or 5% will be entered into cell C9.  To make it easier to reference this value in related formulas the cell is named “'''POPGrow”.''' After the cell has been named it is now possible to refer to it by its name rather than by a cell reference such as “C9”.   This will make the next stage in the process much easier and makes your spreadsheets easier to understand, both for yourself and reviewers.

In the above example (Figure 7) a defined distribution has been used with a low population growth rate of 1% having a 40% probability, medium population growth rate of 3% at 50% probability, and a high population growth rate of 5% at 10% probability.  Each time a simulation is run either 1%, 3%, or 5% will be entered into cell C9.  To make it easier to reference this value in related formulas the cell is named “'''POPGrow”.''' After the cell has been named it is now possible to refer to it by its name rather than by a cell reference such as “C9”.   This will make the next stage in the process much easier and makes your spreadsheets easier to understand, both for yourself and reviewers.