Of course we care about mathematical rigor. > Yes, if you define the rng circularly, then you can make the assumption, We’re agreed then. > but then you cann

Of course we care about mathematical rigor. > Yes, if you define the rng circularly, then you can make the assumption, We’re agreed then. > but then you cannot make the claim of 1/2 per term, since they are not uniform. As mathematicians, we’re allowed to make multi-step arguments. The first step rand(rand()) = rand() * rand() holds for any distribution that scales multiplicatively with the argument, as we’ve agreed. The second step E[rand() * rand()] = E[rand()] * E[rand()] holds for any two independent distributions with finite expectation. The third step E[rand()] = 1/2 holds for this particular distribution. An ideal mathematical rand() satisfies all three of these properties, so this argument that E[rand(rand())] = 1/4 is valid in the mathematical reals. (And it’s approximately valid in the IEEE reals, which is all one typically asks of the IEEE reals.)