RE: Distribution of numbers in the multiplication table
June 8, 2020 at 1:58 pm
(This post was last modified: June 8, 2020 at 2:08 pm by polymath257.)
(June 8, 2020 at 4:53 am)FlatAssembler Wrote: So, what do you guys here think, what would be the distribution of the numbers in the multiplication table? That is, in the 10x10 multiplication table, why are there 6 numbers between 10 and 20 (12, 14, 15, 16, 18), 5 numbers between 20 and 30 (21, 24, 25, 27 and 28), but no numbers between 90 and 100 and only one number between 80 and 90 (81)? I've made a web-app (works only in modern browsers, not even in Internet Explorer 11) that calculates the properties of that distribution, but I can't figure out if it's some distribution that's already been described.
Well, think of it like this.
Suppose you look instead look at products of numbers up to 100. The products can be up to 10,000, but the next highest product is 9900 (twice) and the next below that is 9801 with 9800 (twice) just below that. That means there are six products in the 2% interval from 9800 to 10,000.
So, part of your difficulty is that the 'gap' between factors is 10% of your largest factor. In the case where you multiply up to 100, the gap is a mere 1%.
And, as you increase the largest number available, you get a more and more continuous distribution.
This suggests that we actually want to look at products from the unit interval [0,1] to the unit interval [0,1].
So, what is the probability that a product xy where 0<= x,y <=1 is between the values s and t? This corresponds to the area between the curves y=t/x and y=s/x inside the unit square.
This is actually not too bad to calculate and the answer is (t-s)+s ln(s)-tln(t).
The corresponding distribution function is -ln(t).
This says that products close to 0 are much more likely and those close to 1 much less likely. The distribution itself is fairly well known.
