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**Delimiters and Discrete Distributions**

**Applies to:** @RISK 5.x–7.x

I have a RiskPoisson(3) distribution, and I click Define Distributions, or Browse Results after a simulation. I set the delimiters to 0 and 6, and @RISK shows a probability of 91.7% between them. But Excel's POISSON.DIST(6,3,TRUE) shows a cumulative probability of 96.6%. Which one is right?

This seems strange at first, but there's an explanation. This is nothing special about the Poisson distribution; it applies to RiskBinomial, RiskDiscrete, and all the other discrete distributions.

@RISK and Excel are both right, but they're measuring different things. Excel is reporting the cumulative probability from x=minus infinity to x=6. @RISK reports the cumulative probability from x=0 to x=6. But the Poisson distribution doesn't extend to negative x, so why aren't those two the same?

It's clearer if you look at the cumulative distribution. It's a step function, and x=0 and x=6 are right on the steps. So how is @RISK to allocate the probability for x=0 and the probability for x=6? The answer is that if a delimiter is directly on a discrete x value, @RISK allocates all the probability for that x value to the region to the left of the delimiter. So the 5% probability of x=0 (left-hand delimiter) goes in the left-hand region, and the 5% probability of x=6 (right-hand delimiter) goes into the middle region. The probability shown for the middle region is thus P(L < x ≤ R), not P(L ≤ x ≤ R) as you might expect.

This convention avoids some anomalies. For example, suppose you set both delimiters to 3. If the rule were P(L ≤ x ≤ R), then the middle region, which has zero width, would have a probability of 22.4%, equal to P(x=3), and the visible probabilities would add up to only 77.6% instead of 100%.

Given that this is mathematically valid, it still looks odd at first glance. If you need to make the graph look "right" for a presentation, you can do it easily. Delimiters are rounded to two decimal places, so set them to −0.001 and 6.001. Then x=0 and x=6 will both be inside the center region, but @RISK will display 0.00 and 6.00 for the delimiters.

I clicked in an empty cell, clicked Define Distributions, and selected Poison with λ=3. Initially the graph showed delimiters of 1 and 6, with probability 90% between them. I just clicked on the delimiters, without moving them, and the probability changed to 76.7%. Why?

By default, the Define Distribution graph sets delimiters at the 5th and 95th percentiles. (You can change this default in the Simulation Graph Defaults section of Application Settings.) To show the delimiters, @RISK finds the x values of those percentiles. If you change the percentages, @RISK finds new x values; and if you change the x values, @RISK finds new percentages. When you click on a delimiter, even if you don't actually change it, @RISK takes that as a signal that it should adjust the percentage to the x value instead of the other way around. So it recomputes the percentages based on the x values 1 and 6.

But why should the probability change? Once again, the explanation is in the cumulative graph. One percentile can only be one possible x value, but one x value can be any of a range of percentiles. Thus, computing a percentile from an x value may not give a consistent result with computing an x value from a percentile. This is a feature of any discrete distribution, not just the Poisson.

Last edited: 2015-06-19

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