Understanding “randomness”

Question

I can\'t get my head around this, which is more random?

rand()

OR:

rand() * rand()

I´m f

Answer 1

When in doubt about what will happen to the combinations of your random numbers, you can use the lessons you learned in statistical theory.

In OP's situation he wants to know what's the outcome of X*X = X^2 where X is a random variable distributed along Uniform[0,1]. We'll use the CDF technique since it's just a one-to-one mapping.

Since X ~ Uniform[0,1] it's cdf is: f_X(x) = 1 We want the transformation Y <- X^2 thus y = x^2 Find the inverse x(y): sqrt(y) = x this gives us x as a function of y. Next, find the derivative dx/dy: d/dy (sqrt(y)) = 1/(2 sqrt(y))

The distribution of Y is given as: f_Y(y) = f_X(x(y)) |dx/dy| = 1/(2 sqrt(y))

We're not done yet, we have to get the domain of Y. since 0 <= x < 1, 0 <= x^2 < 1 so Y is in the range [0, 1). If you wanna check if the pdf of Y is indeed a pdf, integrate it over the domain: Integrate 1/(2 sqrt(y)) from 0 to 1 and indeed, it pops up as 1. Also, notice the shape of the said function looks like what belisarious posted.

As for things like X₁ + X₂ + ... + X_n, (where X_i ~ Uniform[0,1]) we can just appeal to the Central Limit Theorem which works for any distribution whose moments exist. This is why the Z-test exists actually.

Other techniques for determining the resulting pdf include the Jacobian transformation (which is the generalized version of the cdf technique) and MGF technique.

EDIT: As a clarification, do note that I'm talking about the distribution of the resulting transformation and not its randomness. That's actually for a separate discussion. Also what I actually derived was for (rand())^2. For rand() * rand() it's much more complicated, which, in any case won't result in a uniform distribution of any sorts.