(Edited because I re-watched the video a bit more closely.) I noticed that Tony wrote "∝n^(-1.3)", i.e. proportional to and not necessarily equal. So let's assume that the probability function f(n)=an^(-1.3) for some constant a. The anti-derivative of that is is F(n)=K-(10/3)an^(-3/10). Because F(n) is the cumulative distribution, the limit as n->infinity of F(n) needs to be 1, so K=1, and we have F(n)=1-(10/3)an^(-3/10).
The other requirement for f(n) to be a probability function is that the area under the curve has to be 1. Let’s find m such that the area from (m,inf) is equal to 1. The area should just be equal to:
F(infinity) - F(m) = 1
1-(10/3)a(infinity)^(-3/10) - (1-(10/3)am^(-3/10)) = 1
1 - 0 - 1 + (10/3)am^(-3/10) = 1
(10/3)am^(-3/10) = 1
(10/3)a = m^(3/10)
a = (3/10)m^(3/10)
Now let's say that we want this distribution to be defined for n=1. Then we want to set m=1 above and solve for a:
a = (3/10)(1)^(3/10)
a = 3/10
And now we can go back to our original probability function definition and say that f(n)=(3/10)n^(-3/10). If you integrate that from 1 to infinity, you'll find that the area is indeed equal to 1. (Never mind that this is a discrete distribution we're talking about, but it's close enough.) That's the part that wasn't covered in the video. The cumulative distribution for that is in fact F(n)=1-n^(-3/10), as shown in the video.