HomeTechniques and Tips@RISK DistributionsCustom Distribution Using RiskCumul

# 3.39. Custom Distribution Using RiskCumul

Note: This article illustrates solutions to very specific problems, but you can modify them to create many different custom distributions.

Example 1:
I need a distribution where there's a 75% chance of a value between 0 and 8 and a 25% chance of a value between minus 12 and minus 7.  A competing product does this as a "custom distribution". Can I do it in @RISK?

Response:
Yes, the RiskCumul function can represent this distribution for you.  In RiskCumul, you specify an array of points and a second array of cumulative probabilities at those points.

Here is the function:
=RiskCumul(minimum, maximum, array of x, array of cum-p)
=RiskCumul(-12, 8, {-7,0}, {0.25,0.25})
Try pasting this formula into an Excel cell and then clicking Define Distribution to see the histogram.

Here's how to read the arguments:

x  cum-pexplanation
−12 0 minimum value of distribution is −12
−7 0.25 25% probability between −12 and −7
0 0.25 0% probability between −7 and 0
8 1 maximum value of distribution is 8

The first two arguments to RiskCumul are the lowest and highest possible values in your distribution. You specified minus 12 and plus 8 in your problem statement.

The array of x's and the array of cum-p's are enclosed in curly braces { }.  (Alternatively, you could put the numbers in cells of your Excel sheet, and then reference the array in the form D1:D4 without braces.)

The 0.25 cumulative probability for x=0 might seem a bit strange. The explanation is that you specified zero probability between minus 7 and 0.  If the probability in that region is zero, then the cumulative probability at every point in the region is the same as the cumulative probability at the left edge, namely 0.25 (25%).

The cumulative probability of 1 is not specified anywhere in the RiskCumul function, because it's implicit in the listing of 8 as the maximum for the distribution.

Example 2:
I need to set up a probability distribution as follows:

• 75% of the probability occurs between 0 and 5
• 25% of the probability occurs between −15 and −5

Response:
Here's how to analyze it:

• The lowest possible value is −15 and the highest possible value is 5.
• The first 25% of cumulative probability occurs between the minimum and −5.
• Values between −5 and 0 are impossible (zero probability), so the cumulative probability remains at 25% at x=0.
• The rest of the probability, 75%, occurs between x=0 and the maximum.

Paste this formula into a cell:
=RiskCumul(-15, 5, {-5,0}, {0.25,0.25})
and press the Enter key.

To see the distribution, click into the cell and click Define Distribution.

RiskCumul takes four arguments: the minimum x, the maximum x, an array of intermediate x's, and an array of the cumulative probabilities for those x's. (Arrays are enclosed in { } curly braces.)  The 75% probability for the region 0 to 5 doesn't appear explicitly — it's implied by the fact that cumulative probability is 0.25 at x=0 and is 1.00 at x=5.

These particular examples show three regions (divided by two x's) between the minimum and maximum, but you could have any number of regions.