# 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.)

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.

• 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 → 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:

• 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

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