Meaning of randomness in space

Question

I am a non-math person and have a question about randomness: When generating elements from a finite set using some algorithm, it is clear what it means when saying that elements should be randomly generated. How about for infinite sets? For example if I want to generate real numbers randomly, what does it mean to be random? In general, how is random defined in Euclidean n-space? How about for subsets of n-space, eg. generating random points on the (n-1) unit sphere? Thanks.

Answer 1

There are two aspects: First, you need to specify a probability distribution. This is a function that says for a subset (ignoring questions of measureability for the moment) how probable it is for a random element to lie in this set.

Example: You have a finite set $$ A = \{1, 2, 3, 4 ,5 , 6 \} $$ A probability distribution $p$ would be: $$ p(x) = \frac{1}{6} $$ for $x = 1, ..., 6$. Note that the sum over all probabilities needs to be $1$.

For an infinite set like the interval [0, 1], a probability distribution would be a function that has an integral of 1: $$ \int_{0}^1 p(x) d x = 1 $$ The simplest possible example would be the uniform distribution, the constant function $$ p(x) = 1 $$ (This time I implicitly assumed that we are talking about probability distributions that have a Radon-Nikodym density with respect to the Lesbegue measure.)

On "infinite" sets like the whole real line (or $\mathbb{R}^n$), there is no uniform distribution, because its integral could not be 1 anymore, but there are a lot of interesting other distributions.

The second important aspect is independence: You need to specify if you know anything about the next random element if I tell you about what happended before. If you say: What happens next, i.e. the next random element, is independent of what happened before, one says that the random elements are independent. But there are a lot of interesting situations where this is not so. When you play in a casino and have a fixed budget, you cannot play if you go bankrupt, for example.- In this case "what happens next" does depend on the events that happended before.

To pick up one of your examples: The unit sphere in $\mathbb{R}^n$ has finite Lesbegue measure, so that it is possible to define the uniform probability distribution on it. And lets also say that we would like to generate elements that are independent. In the one dimensional case, i.e. the unit circle, we could generate independent uniformly distributed elements of the unit interval $ x \in [0, 1]$ and calculate $$ e^{i x} = \cos(x) + i \sin(x) $$ which will result in numbers that are uniformly distributed on the circle. (In case you don't know about complex numbers, you can write the latter as $(\cos(x), \sin(x))$, in cartesian coordinates in $\mathbb{R}^2$).