# 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 ] [ https://en.wikipedia.org/wiki/Randomness ]

[ Random variable ] [ https://en.wikipedia.org/wiki/Random_variable ]. → A variable with random values. aka stochastic variable.

[ Random process ] [ https://en.wikipedia.org/wiki/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.

• a probability function for discrete random variable is called probability mass function
• a probability function for continuous random variable is called probability density function

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

[ Probability distribution ] [ https://en.wikipedia.org/wiki/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 ] [ https://en.wikipedia.org/wiki/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:

• Define a domain of possible inputs.
• Generate inputs randomly from a probability distribution over the domain.
• Perform a deterministic computation on the inputs.
• Aggregate the results.

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:

• Draw a square on the ground, then inscribe a circle within it.
• Uniformly scatter some objects of uniform size (grains of rice or sand) over the square.
• Count the number of objects inside the circle and the total number of objects.
• The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4. Multiply the result by 4 to estimate π.

## Entrophy

[ Entropy (information theory) ] [ https://en.wikipedia.org/wiki/Entropy_%28information_theory%29 ]

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