# What is Random? Random is modeled in mathematics by defining a set of possible events that can occur and count the actual ratio each occurring event and the number of events. The ratio is called probability and paired with the events they define a distribution. This is the basic model of randomness. The sum of all probabilities is 1. Uniform distribution describe maximum randomness where each outcome has the same chance to occur. It has the maximum possible [[Entropy|entropy]] among all distributions on $n$ outcomes. It is fundamental in the sense that all other distributions can be generated from it. There are multiple ways to convert uniform to non-uniform distributions: [[Discrete sampling from uniform sampling with Knuth-Yao algorithm|Knuth-Yao]] and [[Rejection sampling]] for general discrete distributions, [[Box-Muller]] for normal distributions. Random is useful. It is useful in clinical trials where patients have different genetics and habits that can effect the trial result. Taking a random group of people helps as randomization makes these factors statistically similar across groups, so the average difference approximates the treatment effect. Random is used to approximate solution for certain complex math problems. Problems that can be described as area or as integrals with [[Monte Carlo methods]]. It can help validate hard-to-compute-criteria over large spaces like in the case of [[Miller-Rabin primality test]] and [[Pollard's Rho factoring]]. Deciding if data is truly random or only looks random but actually well structured - can be [[Hard computation problems|computation hard]]. For example the [[LWE problem|LWE]] problem. This phenomena is the basis of multiple cryptography schemes and specifically [[CKKS scheme (index)|FHE]]. More generally, hiding information by adding data that looks like random that can be removed using some secret. Accidents happens. Knowing that events are accidental (random) can remove concerns that something deliberate happened and so might be prevented. Are the [[The Prussian cavalries (Poisson distribution)|death by horse kicks]] accidental and can't be avoided or not, are the locations of [[Bombing London (Poisson distribution)|falling bombs on London during WWII]] random or deliberate? Quantum physics is heavily modeled on probability theory. A key question: does true random exist? For example, is a sequence of bits random? The binary string problem lead to the definition of [[Algorithmic Randomness]] that use the notion of incompressibility. Another definition is to measure how "close" are its substrings to uniform distribution by computing their [[Measure distribution randomness with Entropy|entropy]] and another using the [[Null Hypothesis]]. Pseudo-random generators are algorithms that generate randomness using a computer although it is a deterministic device. Computers store random events and then generate high quality uniform sampling. This storage is called the OS entropy-pool. An interesting fact is that randomness can have structure. The average of random events (sampled from any distribution even from uniform distribution) have normal distribution (order) - this is the [[Central limit theorem (CLT)]] . A balance between random and order enable the creation and stability of complex structure in nature like the DNA, living cells animals and even society and culture. Randomness has crucial part in [[Randomness is important to life|supporting life]]. The study of randomness started in the context of gambling by asking which events are more probable than others. This helps planning how to win games like blackjack and the roulette. This lead to the definition of the concept of the distributions: [[Bernoulli Trial Randomness|Bernoulli]], Uniform and [[Poisson distribution|Poisson]]. High level order can emerge from low level randomness. In mathematics, the average of samples from uniform distribution (random) have normal distribution (order). In physics, random motion of particles is described by exact formula for their pressure and temperature (order). Randomness behavior can be counter intuitive as can be seen by [[Monty Hall paradox]] and [[Simpson paradox]]. Randomness is different from chaos. chaos means there are fundamentally no rules to describe it. It is indescribable. No language can catch it. In western thought, order is assumed. The physical events are controlled by physics rules based on mathematics. Therefore, chaos can't exist. Random events, although unpredictable, are controlled by rules of mathematics. ## Questions - Quantum mechanics physics is ruled by randomness as described by quantum mechanics. how this relates to randomness? - How sampling from a distribution is related to randomness? Reference:[[@What is Random? Chance and Order in Mathematics and Life]] ## Created 2024-06-12 22:07