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**Specifying Distributions in Terms of Percentiles**

**Applies to:** @RISK 5.x–7.x

I know what distribution I want to use, but I want to specify it in terms of percentiles rather than with the usual parameters. Is there a way?

For many distributions, you can. We use the term "Alternate Parameters" for specifying at least one percentile in place of a usual parameter like mean, most likely, alpha, and so forth.

In the Define Distributions dialog, select the *Alt. Parameters* tab, and you'll see the distributions that can be specified in terms of percentiles. Double-click your desired distribution to select it.

A dialog will open with some suggested percentiles, and you can change the values in that dialog as usual. But quite possibly you'll want to specify different percentiles from the suggested ones — for example, the 10th and 90th instead of the 5th and 95th. To change which percentiles are used, click the drop-down arrow at the right of the word *Alternate* to open the Parameters dialog. If you open the Parameters dialog, you can change which percentiles are used, and by selecting the radio buttons you can even define the distribution based on a mix of percentiles and standard parameters. For more, with a screen shot, please search for *Alternate Parameters* in @RISK's help file.

How do percentile parameters work internally? Does @RISK convert them to standard parameters?

Yes, @RISK resolves percentile parameters into standard parameters. This has to be done in every iteration in a simulation, because it's possible for your workbook's logic to change the parameters of a distribution from one iteration to the next.

In general terms, resolving alternate parameters is a kind of optimization problem. Say you have a potential candidate for the resolved (non-Alt) distribution. You can calculate an error for this candidate by computing the difference between the desired percentiles specified in the Alt function, and the percentile values your candidate actually has. Finding the correct non-Alt distribution requires juggling the parameters until that error goes to zero. That's the simplest method, and indeed you could use Palisade's Evolver or RISKOptimizer, or Excel Solver, to resolve alternate parameters yourself using this method.

But there's a problem with this method: it's just not fast enough, especially for complicated cases like the BetaGeneralAlt with its four parameters. The time to solve an optimization problem goes way up as you increase the dimensionality. If you were resolving parameters just one time, it would probably be fine to do it in this brute-force way; but, given the possibility of different parameters in each iteration, the resolution process needs to be really fast. Fortunately, we can usually reduce the dimensionality of the optimization. For example, with BetaGeneralAlt, some tricky math reduces the problem from four dimensions to two. (The differing mathematical tricks for each distribution are proprietary. Our developers put a lot of work into making them as efficient as possible.)

Can I see the standard parameters that @RISK computes from the percentiles?

You can see approximate values, rounded to a few decimal places. In the Define Distributions dialog, above the parameters click the drop-down arrow to the right of *Alternate*. On the Parameters dialog that opens up, remove the check mark next to *Alternate Parameters*. @RISK will then display the non-Alt distribution that matches the Alt distribution as closely as possible.

For example, paste this formula into an empty cell:

=RiskBetaGeneralAlt(5%,-3,25%,2,75%,12,95%,16)

Press Enter, and then click Define Distributions. Click the drop-down arrow to the right of *Alternate*, remove the check mark for *Alternate Parameters*, and click OK just once. The display now shows the equivalent regular parameters, σ1=1.295682, σ2=1.121222, Min=-4.990886, Max=17.204968. (Because these are rounded values, some statistics and percentiles may be slightly different from their values in the Alt distribution.) The full non-Alt distribution is shown in the *Cell Formula* box near the top of the Define Distributions dialog:

=RiskBetaGeneral(1.295682,1.121222,-4.990886,17.204968)

If you now click OK, @RISK will replace the Alt distribution in your worksheet with that non-Alt distribution; if you click Cancel, the Alt distribution will remain in your worksheet.

Last edited: 2015-12-01

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