Not trying to be snarky by any means. I'm sorry if you interpreted it that way.

I don't think the difference will show up for small N. This is an asymptotes thing. Try it for N = 100, that's what I did. For example:

  >>> np.product(np.random.random(100))
  5.469939152265385e-43

  >>> 1 / np.e**100
  3.7200759760208555e-44

  >>> 1 / 2**100
  7.888609052210118e-31

The underlying thing here is that random(random()) in this case is the same as random() * random(). So random(random(random(...))) is the same as random() * random() * random() and then the analysis goes on. And sure, random() * random() has a mean close to 1/4. But the dynamics change as N becomes large.

Edit - and just in case you doubt whether random() * random() * ... is a valid way of doing this, I also just checked the laborious way of doing it:

  >>> def foo(n):
  ...   result = 1.0
  ...   for _ in range(n):
  ...     result = np.random.uniform(0, result)
  ...   return result
  ...
  >>> foo(100)
  1.4267531652344414e-46
  >>> foo(100)
  7.852496730908136e-49
  >>> foo(100)
  1.3216070221780724e-41