Random Notes on Randomness

By Xah Lee. Date:

random learning notes. (I studied probability, statistics, game theory, back in ~1993. Here's a refresher for myself.)

Randomness

Random variable. → A variable with random values. aka stochastic variable.

Random process aka “stochastic process” is a collection of random variables.

random variable can be discrete (such as any integer from 1 to 10), or continuous (such as any real number between 1 to 10)

probality function → a function that takes a value (of possible values of a random variable) and returns its probability.

probability is a number between 0 and 1, inclusive. 0 means never gonna happen. 1 means sure thing.

Probability distribution → the possible values of a random variable and their associated probabilities.

a probability distribution can be for one single random variable (this is most common), or multiple random variables. When it's single variable, it's called univariate. For multiple variable, it's multivariate.

Monte Carlo method basically, a method to test some theory or answer some question by simulating the event using random data. « Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results; i.e., by running simulations many times over in order to calculate those same probabilities heuristically just like actually playing and recording your results in a real casino situation: hence the name.»

Monte Carlo methods vary, but tend to follow a particular pattern:

For example, consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is π/4, the value of π can be approximated using a Monte Carlo method:

Entrophy

Entropy (information theory)

PARADOXES OF RANDOMNESS By Gregory Chaitin. (Complexity, Vol 7, No 5, May/June 2002, Pp 14-21) At http://www.cs.auckland.ac.nz/~chaitin/summer.html

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