Applies to: @RISK 5.x–and Up

 

The @RISK Poisson Distribution vs. Excel’s POISSON.DIST

Issue
The probability shown by @RiskPoisson(3) does not match Excel’s
POISSON.DIST(6, 3, TRUE).

At first glance this appears to be an error, but both @RISK and Excel are correct—they are simply measuring cumulative probability differently.

Why the Probabilities Differ

Excel reports the cumulative probability:

P(x ≤ 6)
(from minus infinity to 6)

@RISK, however, reports cumulative probability over an interval defined by graph delimiters:

P(L < x ≤ R)

By default, @RISK places delimiters at the 5th and 95th percentiles and displays the probability between them.

Why This Matters for Discrete Distributions

This behavior is not specific to the Poisson distribution—it applies to all discrete distributions, including:

• RiskPoisson

• RiskBinomial

• RiskDiscrete

• and others

Discrete distributions have step-function cumulative distributions. When a delimiter lies exactly on a discrete x-value, @RISK must decide how to allocate the probability mass at that point.

@RISK Convention

If a delimiter lies exactly on a discrete value:

• All probability at that x-value is assigned to the region to the left

• The middle region therefore represents:
  P(L < x ≤ R), not P(L ≤ x ≤ R)

Example:

• Probability of x = 0 goes to the left region

• Probability of x = 6 goes into the middle region

• 

   

  

Why @RISK Uses This Rule

This convention avoids logical inconsistencies.

For example, if both delimiters were set to 3:

• The middle region would have zero width

• If the rule were P(L ≤ x ≤ R), the middle region would still show P(x = 3)

• The visible probabilities would no longer sum to 100%

@RISK’s approach guarantees that probabilities always add up correctly.

Making the Graph Look “Right” for Presentations

If you want x = 0 and x = 6 visually included in the middle region:

1. Set delimiters to −0.001 and 6.001

2. @RISK rounds displayed delimiters to two decimals

3. The graph will show 0.00 and 6.00, but include both values in the center region

Why Clicking a Delimiter Changes the Probability

Scenario

• You define RiskPoisson(3)

• The graph initially shows delimiters at 1 and 6

• The probability between them is 90%

• 

• You click on a delimiter (without moving it)

• The probability changes to 76.7%

• 

Explanation

By default:

• @RISK sets delimiters based on percentiles (5% and 95%)

• It computes the corresponding x-values

When you click a delimiter, @RISK interprets this as:

“Use these x-values to recompute the percentiles.”

Because discrete distributions map:

• One percentile → one x value

• One x value → a range of percentiles

The reverse calculation does not necessarily give the same result.

This is normal behavior for any discrete distribution, not just Poisson.

Summary

• Excel and @RISK use different—but valid—definitions of cumulative probability

• @RISK’s delimiter rules are necessary for consistency in discrete distributions

• Clicking a delimiter switches the calculation direction (percentile → x or x → percentile)

• Apparent discrepancies are expected and mathematically correct