What is Random: Definition and 1000 Discussions

In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.
The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.
Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, particularly in the field of computational science. By analogy, quasi-Monte Carlo methods use quasi-random number generators.
Random selection, when narrowly associated with a simple random sample, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random.According to Ramsey theory, pure randomness is impossible, especially for large structures. Mathematician Theodore Motzkin suggested that "while disorder is more probable in general, complete disorder is impossible". Misunderstanding this can lead to numerous conspiracy theories. Cristian S. Calude stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness". It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.

View More On Wikipedia.org
Recent contents Top users
  1. dRic2

    Calculating the Probability of Two Men Meeting Again with a Double Random Walk

    I set up the problem in the following way: considering the relative motion, at each step there is a probability that - they take a step in the opposite direction going away from each other, so the distance increases and the associated probability is 1/4 - they take a step in the opposite...
  2. E

    B How do we interpret a random variable?

    I've read that we can define a random variable on a probability space ##(\Omega, F, P)## such that it is a function that maps elements of the sample space to a measurable space - for instance, the reals - i.e. ##X: \Omega \rightarrow \mathbb{R}##. That being said, it's often treated (at least...
  3. J

    MHB Minimum Degree of a Random Graph (Probabilistic Method)

    Problem: Suppose that the function $p : N \rightarrow [0, 1]$ satisfies $p >> n^{-1}ln(n)$ (i.e. $n^{-1}ln(n) = o(p)$). (a) Prove that as $n \rightarrow \infty$, the random graph $G(n, p)$ has minimum degree at least $\frac{np}{2}$ almost surely. Idea: Look at the degree of each individual...
  4. Chris Miller

    B Random lengths of Collatz chains

    Assuming its hail stone series is pseudo-random, the Collatz algorithm divides by 2 twice for each multiplication by 3. This means that for every three iterations the test value is on average reduced by 1/4, or 1/12 per iteration. I've tweaked the algorithm to produce much longer series for...
  5. J

    MHB Are Events in Random Graphs from G(n, 1/2) Independent?

    Let $n$ be a positive integer, and let $G$ be a random graph from $G(n, 1/2)$. Let $e_1, . . . , e_{n \choose 2}$ be the possible edges on the vertex set ${1, . . . , n}$, and for each $i$, let $A_i$ be the event that $e_i ∈ E(G)$. Prove that the events $A_1, . . . , A_{n \choose 2}$ are...
  6. SchroedingersLion

    A Langevin equation - derivative of random force?

    Greetings, I am struggling with an exercise to the Langevin equation. Suppose we are given the following differential equation for a particle's 1D time-dependent momentum ##p(t)##: $$\text{d}p = -\gamma p \text{d}t + F(r)\text{d}t + \sqrt{C\gamma}\text{d}W $$ with a constant ##C##, a...
  7. A

    A Third and fourth central moment of a random variable

    My question is as follows. In the attached paper a formula is given on page 272 for the expectation of Tn (formula 23) and for the variance of Tn (formula 24). Now I would like to know what the formulas look like for Tn 's third and fourth central moment.
  8. L

    Comp Sci C++ random number generation

    The program works as intended. #include <iostream> #include <ctime> #include <cstdlib> int coinToss ();int main () { int tosses, result; std::cout << "How many coin tosses? "; std::cin >> tosses; size_t seeder = time(0); srand(seeder); for (int i = 0; i < tosses...
  9. F

    I Distribution of a sample random variable

    $X_1, X_2, ..., X_{15}$ are independently to each other and follow $N (7, 3^2)$ what distribution the following statistics follow$T = \frac{(\bar{X}− 7)}{\sqrt{s^2/15}}$i know this follow t distribution $t_(n-1) =t_{14}$but how do i find what distribution $T^2$ follows, can i just multiply it?$T...
  10. H

    Unit of Velocity from a Random Walk measured by an accelerometer

    Hi, I am working on a kalman filter where my measurement equation involves "-g + v" , where g is in m/s^2 and v is velocity random walk given in m/s/sqrt(hr). Feels like a stupid question, but how can I transform the unit of velocity random walk so I can do the calculation correctly?
  11. Biochemgirl2002

    Calculating general normal random probability

    a) P(X<18) = (18-20)/sqrt25 =-2/5 =-0.4 then you use the standard normal table and find that; P(X<18)=0.3446 b) P(X>27) = (27-20)/5 = 7/5 = 1.4 P(Z>1.4) =P(Z<-1.4) =0.0808 C) =(13<X<23) =13-20/5 , 23-20/5 =-7/5 , 3/5 =-1.4 , 0.6 P(Z<0.6)-P(Z<-1.4) =0.7257-0.0808 =0.6449
  12. S

    Identifying and understanding random number algorithms

    Studying different ways to generate random numbers according to a distribution and the below algorithm describes the "box method". A search on Google led to the Box-Mueller method. Are they related? Also, what would be a simple implementation of this algorithm for ##f(x)=\sin{(x)}## on...
  13. I

    Optimized Algorithm to Find All 720 Random Permutations of [0, 6)

    Summary: Algorithm do find all random permutations of n=[0,6) Hi, The following algorithm gives 6 out of the 720 random permutations of integers in the range [0, 6). t=0..5 // permutation i=0..5 // sequence index n =...
  14. kyphysics

    Random Rounding in Google Sheets

    If anyone knows what on Earth could be happening and how to fix this, please let me know. I'm using Google Sheets. I have lots of cells where I'm manually entering numbers with some having decimal places. All of a sudden, a random cell rounds a number UP for me. WTH? There's no rounding...
  15. F

    A Understanding Kolmogorov-Smirnov Test Results for Random Number Generators

    Hi all, I have some doubts regarding the Kolmogorov test: I made a simple c++ program generating two samples of random numbers following a Landau distribution (I used the "hit and miss" method). I made the Kolmogorov test, in order to check the randomness of the generator, but I'm having some...
  16. M

    MHB Cdf, expectation, and variance of a random continuous variable

    Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere. 1. What is the value of b such that f(x) becomes a valid density function 2. What is the cumulative distribution function F(x) of f(x) 3. What is the Expectation of X, E[X] 4. What is...
  17. M

    How to program random processes

    A simple example would be to take a deck of 52 cards and deal them out until the ace of spades is seen except that I'd quit after I saw 20 cards. I'm not very sophisticated in programming. I use GW Basic, which has been generally adequate, and I have much experience with Excel and Lotus...
  18. Spinnor

    B Can you pick out the random set of points in a plane?

    The above screen shot is from, @ the 1:43 mark. Which set of points is random? Answer below. The right set of points.
  19. O

    I What Are the Odds that Two Random Events Occur at the Same Time?

    Suppose someone randomly went to Starbucks Monday thru Saturday between 8:00 AM and 10:AM. Suppose one Tuesday in May they purchased a drink and the credit card transaction occurred at 9:14:12 AM. What are the odds that person could randomly return on a Wednesday in July and purchase a drink...
  20. R

    Can a random sequence produce an ordered output

    Is it possible for order to arrive out of disorder on a macro scale? Contrary to the 2nd law? Specifically, is it scientifically acceptable to believe that the random mutation of genetic material, which was itself produced by the random coupling of molecules has resulted, over the extended...
  21. WMDhamnekar

    MHB Transformation of random variable

    Hello, A discrete random variable X takes values $x_1,...,x_n$ each with probability $\frac1n$. Let Y=g(X) where g is an arbitrary real-valued function. I want to express the probability function of Y(pY(y)=P{Y=y}) in terms of g and the $x_i$ How can I answer this question? If any member...
  22. arcTomato

    Solution to Random Walk Problem | n_i

    I don't have any idea to prove this 😢 ##n_i##is number of trial, right?
  23. arcTomato

    Please teach me how to prove this random walk problem....

    I think I can prove equal length version problem, But I am confusing in this case,,, Please help me!
  24. F

    A Random Quantum Walk: Learn & Use w/ Quantum Gates

    I am an undergraduate doing research on QC/QI. My current topic to learn is continuous-time quantum walks, but first I must learn the random quantum walk. That being said, I was wondering if someone could simply explain what a random quantum walk is and then explain how they could be useful with...
  25. L

    I A random question comes to mind, about the infinitesimal area of rings

    I know the area of a thin ring of radius ##r## can be expressed as ##2\pi rdr##, however, I wonder if I use the usual way of calculating area of a ring, can I reach the same conclusion? I got this: $$4\pi(r+dr)^2-4\pi r^2=4\pi r^2+8\pi rdr+4\pi (dr)^2-4\pi r^2=8\pi rdr+4\pi (dr)^2$$And now I'm...
  26. C

    MATLAB Problem with random variables in Matlab's PCA

    Hello. I have designed a Gaussian kernel as: [X,Y] = meshgrid(0:0.002:1,0:0.002:1); Z=exp((-1)*abs(X-Y)); Now, I calculate PCA: [coeffG, scoreG, latentG, tsquaredG, explainedG, muG]=pca(Z, 'Centered',false); I can rebuid the original data propperly as defined in the dcumentation...
  27. S

    Modes and Q-factor(s) of a random laser

    Many papers about random lasers mention the Q-factor of random lasers. Since a random laser has multiple peaks close to each other like shown in the figure. Does each of these peaks correspond to a unique random lasing mode, or is it just a single mode? Similarly what is the right way to...
  28. R

    I Sum of many random signals

    For instance, here is an example from my own simulations where all underlying signals follow the same analytical law, but they have random phases and amplitudes (such that the sum of the set is 1). The thick line represents the sum: Clearly, the sum tends to progressively get flatter as ##N...
  29. WMDhamnekar

    MHB Distribution and Density functions of maximum of random variables

    1] Let X,Y,Z be independent, identically distributed random variables, each with density $f(x)=6x^5$ for $0\leq x\leq 1,$ and 0 elsewhere. How to find the distributon and density functions of the maximum of X,Y,Z.2]Let X and Y be independent random variables, each with density $e^{-x},x\geq...
  30. R

    I Re-scaling of exponentially distributed numbers

    For simplicity, let ##N=1##. The following histograms show my results. The generated random numbers are initially exponentially distributed. But after re-scaling they become almost uniformly distributed. What is the cause of that, and is there a solution? P.S. Here is my code in Matlab...
  31. jaumzaum

    I Stratified Random Sampling vs Simple Random Sampling

    I'm beginning to learn statistics and I quite didn't understand the formula for the stratified random sampling. Let's say we have a country with 3000 people, divided into 3 cities containing 1000 people. We want to know the proportion of women in the whole country so we decide to take a sample...
  32. S

    MHB Probability of a random subset of Z

    I'm stuck in this question, could someone give me a hand? Question 9: Let A = (1,2,3,4) and Z = (1,2,3,4,5,6,7,8,9,10), if a subset B of Z is selected by chance calculate the probability of: a) P (B⊂A) B is a proper subset of A b) P (A∩B = Ø) A intersection B =empty set Appreciate
  33. A

    A Can we create a random variable using QED effects?

    Quantum Electrodynamics (QED) has some observable effects such as the lamb shift, which is mainly caused by the vacuum polarization and the electron self-energy. These effects contribute to the "smearing" of the electron in an unpredictable manner, other than the uncertainty we already have...
  34. user366312

    A Question about the Poisson distributed of random variable

    The following is related to Poisson process: $$P(N_1=2, N_4=6) = P(N_1=2, N_4-N_1=4) = P(N_1=2) \cdot P(N_3=4)$$ Why is $$(N_3=4)=(N_4-N_1=4)$$? Can anyone explain?
  35. Lunct

    Is Brownian Motion actually random or does it follow determinism?

    If you have enough information could you not determine with certainty the movements of pollen particles in water? In other words, if you were able to measure the movements the particles, then repeat the exact same experiment, with all things controlled, would the particles move in the same way...
  36. Boltzman Oscillation

    How can I determine the random variables for this problem?

    So i first need to come up with the sample space, X, and Y. Well I would guess that the random variables here are N1 and N2 and thus X = N1 and Y = N2. Now i need to make these random variables a function of L. I don't know what L should be but I would guess it is the outcome of a 1ms interval...
  37. WMDhamnekar

    MHB Check Martingale Sequences from i.i.d. Variables | Stats SE

    How to answer this question $\rightarrow$https://stats.stackexchange.com/q/398321/72126
  38. DuckAmuck

    I Random Unit Vector Angle Difference

    I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles. In math speek: Ai = uniform(0,2π) dA = Ai - Aj dA is not uniform. Here is a rough image of...
  39. Peter Morgan

    A The relationship between random fields and quantum fields

    My paper "Classical states, quantum field measurement", arXiv:1709.06711, has been accepted by Physica Scripta, https://doi.org/10.1088/1402-4896/ab0c53. The version as submitted to Physica Scripta on November 4th, 2018 is available as arXiv:1709.06711v5. I believe that anyone who puts some...
  40. DaveC426913

    B What does this mysterious formula found in a financial institution mean?

    Found this formula on a whiteboard. I work in a financial institution. Can this mean something? (I may have transcribed the handwriting poorly. The d's might be a's.)
  41. R

    Can random, unguided processes produce a rational brain?

    I am fascinated by Einstein’s quote that the most unbelievable aspect of the universe was that it was intelligible. So my question is does anyone know whether it is so unlikely as to be absurd to suppose that random unguided processes could produce a rational brain in man in as little as 3...
  42. F

    B What's the meaning of "random" in Mathematics?

    Physics, Economists, Biologists, Astronomers and my brother all love the word "Random", as that allows allows them to get out of clockwork processes and allow for variations due to unknowns or whatever else. But, how does a Mathematician reconcile itself with the idea of random? There's no...
  43. J

    MHB Probability of 4 Adults Having Health Insurance

    66% of adults have health insurance, what is the probability that 4 adults selected at random have health insurance?
  44. S

    MHB Probability of Missed Bullseye 3+ Times in 4 Shots

    1.Mary has two friends, Ann and Sarah. Mary will visit Ann this evening if the 19A bus arrives before 6PM. Otherwise, She will visit Sarah. The probability of the 19A bus arriving before 6pm is 40%. If she visits Ann, the probability that Ann will be at home is 10% and if she visits Sarah, the...
  45. Kitty123

    Map a one dimensional random walk to a two-state paramagnet

    1. The question asks us to map a one dimensional random walk to a two state paramagnet and then write an expression for the number of journeys of N steps which end up at r=Rdelta. Then we are asked to find an expression for the probability that N steps will end up at r. 2...
  46. WMDhamnekar

    MHB Joint probability distribution of functions of random variables

    If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation. Hint:The joint density function can be obtained by differentiating the...
  47. WMDhamnekar

    MHB Two normal independent random variables

    Let X and Y be independent normal random variables each having parameters $\mu$ and $\sigma^2$. I want to show that X+Y is independet of X-Y without using Jacobian transformation. Hint given by author:- Find their joint moment generating functions. Answer: Now Joint MGf of...
  48. JesW87

    B Is radioactive decay truly random?

    Before you report this, yes I do know there was already another post like this one, but I don't feel like it fully answered the question. Note that I really don't know anything about quantum anything, but I'm trying to do some reading up on "randomness" and the consensus seems to be that this...
  49. entropy1

    B Can this be called 'a coincidence'?

    Suppose we have two variables A and B. A has a truly random distribution over {0,1} with P(0)=P(1)=0.5 . B has the same distribution. Now suppose that A and B always show both a 1 or both a 0. This would be a strong correlation between A and B. Now could this be called 'a coincidence'? And if...
  50. D

    A Probability of finding a k-subset in a d-dimensional random

    For a somwehat simple version of the problem, imagine a group of ten people (n=10), for which we observe two binary attributes (d= 2). One of the attributes tells us if any given person wears glasses (d_1) the other if any given person wears a hat (d_2). We know that in total 5 people wear hats...
Back
Top