Why is the distribution of the sum of the values on two dice bell-shaped and symmetric if two uniform dist. sum is triangular distribution via Irwin-hall distribution?
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4$\begingroup$ Hm. It isn't, it's triangular. Try a simulation, tabulate the results, and plot a histogram. $\endgroup$– Stephan KolassaCommented Jul 25, 2023 at 15:34
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2$\begingroup$ This exact question (at least your first sentence) is answered in a recent 3 Blue 1 Brown video. $\endgroup$– AlexisCommented Jul 25, 2023 at 16:19
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2$\begingroup$ @jkj What leads you to the premise that it's bell shaped? (I presume you read that claim somewhere, but it's incorrect; it might be worth identifying where the notion came from in the question). The distribution is discrete triangular; it's easy to compute it and see that the left half increases linearly and the right half decreases linearly (and symmetrically to the left half) $\endgroup$– Glen_bCommented Jul 25, 2023 at 17:14
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1 Answer
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About 50 years ago, I did a simulation of this for a middle school science project. This was before computers, so I had to roll the dice by hand and tabulate by hand as well. Then I compared my results to the results predicted by the laws of probability.
They were very close!
The shape is triangular. Only with a low N would it look like a bell. And it's pretty straightforward to show the results (I will leave the answers in /36 for clarity):
- 2 or 12 ... 1/36 (each)
- 3 or 11 .... 2/36
- 4 or 10 .... 3/36
- 5 or 9 .... 4/36
- 6 or 8 .... 5/36
- 7 ........6/36
I didn't win. Some kid who showed pictures of his molting snake won. One of the teachers told me they didn't understand my project.
You can try simulating it yourself, but I recommend using a computer - rolling dice is BORING.
answered Jul 25, 2023 at 16:26