Simulating a random outcome

Known:

There are 256 events. Each event can have a certain number of outcomes of 0 or more. This is no defined upper limit. The average number of outcomes per 256 events is ~2.8945.

Maximum outcome detected in a single event is 7 but could theoretically be higher.

Here are actual results based on a data set of 16 events. This particular sequence happens to be below average. (1.75.)

3, 1, 0, 2, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 1, 3

Based on this known information, how to calculate/simulate using RNG a typical result?

I have several outcome types with different averages and max values and would like to apply this approach to all in a simulation.

EDIT:

I have tried this formula I found somewhere:

$AVERAGE +$STD * sqrt(-2 * log rand) * cos(2 * $PI * rand)  And get this output when applied randomly: 4 3 3 2 3 4 1 3 5 0 2 2 3 3 2 1$AVERAGE = 2.8945
\$STD = 1.75
Run Avg = 2.5625


1.75 for "STD" is arbitrary and something I played with when testing. This seems to be fairly close to what I want.

• There's not enough information to get at the distribution of outcomes per event. One might guess Poisson, but that can't be confirmed or denied based on the very short sample we have. – Brian Tung Apr 16 '18 at 19:18
• @BrianTung What if I added assumption that results are not evenly distributed but conform to a normal distribution pattern similar to a D6 + D6 table? Even though I don't exactly know where the peak might be. I think somewhere between 1-2. I don't actually know all the outcomes but reasonable guesses are okay. – Timothy B. Apr 16 '18 at 19:29
• If there's more to this problem than you're presenting here, now would be the time to give it. As I read it now, there's not enough information, but your comment sounds as though you have additional context that you may not think is relevant. It's hard to tell if we don't see it! :-) – Brian Tung Apr 19 '18 at 0:28
• @BrianTung One big problem I see is that I didn't provide MIN, MAX, MEAN, etc. I could collect that data but it would be high effort. I have several types of events, each with different parameters. I'll edit my question to show what I've tried. – Timothy B. Apr 19 '18 at 15:46