# What is Random? Random is fascinating: total disorder, absolute chaos, no structure. Hard to imagine what this means. It is weird that absolute disorder is treated by the most ordered and structured tool exist - mathematics. Uniform distribution describe maximum randomness. 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. Deciding if data is random or well structured can be [[Hard computation problems|computation hard]]. This is the [[LWE problem|LWE]] problem and it is the basis of multiple cryptography schemes and specifically [[CKKS scheme (index)|FHE]]. More generally, random is used in cryptography to hide information. because adding information to random string remains a random string. Random is useful. It is also useful in clinical trials where patients have unique different genetics, 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. It 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]]. A key question: does true random exist? Is an event random? For example, is a sequence of bits random, the number of [[The Prussian cavalries (Poisson distribution)|death by horse kicks]], the distribution of [[Bombing London (Poisson distribution)|bombing London]] - are those events random or not? The binary string problem lead to the definition of [[Algorithmic Randomness]] that define randomness with incompressibility. Another definition is to measure how "close" its substrings to uniform distribution by computing the [[Measure distribution randomness with Entropy|entropy]] of the string and by the [[Null Hypothesis]]. Pseudo-random generators are algorithms that generate randomness using a computer although it is a deterministic device. Uniform sampling is computed by reading the OS entropy-pool which is a store of noisy event collected and mixed by the OS. 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]]. ## 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