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.

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  1. shivajikobardan

    I want to make a Nepali Random Name Generator

    What's the full process for it? Don't I need an API for names? We don't have that yet. So, how will I do this? Can you guide me step by step? without telling me any code? I'm planning to do this in javascript so that I can deploy the application on the web(blogger).
  2. C

    Uncertainty: Systematic & Random

    I am thinking that it might could be absolute uncertainty = systematic uncertainty + random uncertainty. Many thanks!
  3. WMDhamnekar

    I Expected number of random variables that must be observed

    In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ## In answer to (b), XN is wrong. It should be XN=p-4q-3 - p-3 q-2- p-2 q-1 - p-1. This might be a typographical error. Is my answer to (a) correct?
  4. F

    I Sample space, outcome, event, random variable, probability...

    Hello, I am solid on the following concepts but less certain on the correct understanding of what a random variable is... Random Experiment: an experiment that has an uncertain outcome. Trials: how many times we sequentially repeat a random experiment. Sample space ##S##: the set of ALL...
  5. F

    I Linear regression and random variables

    Hello, I have a question about linear regression models and correlation. My understanding is that our finite set of data ##(x,y)## represents a random sample from a much larger population. Each pair is an observation in the sample. We find, using OLS, the best fit line and its coefficients and...
  6. S

    I Experiment with gamers, random numbers and entanglement

    https://www.wired.com/story/this-random-video-game-powers-quantum-entanglement-experiments/ I don't understand the principle of this experiment. The gamers produced random numbers, and what was done with these numbers then? Was the value like <S> in CHSH inequalities computed, and was it...
  7. physicsclaus

    How to calculate gating time from the rate of the random coincidence?

    Hello everyone, I am now doing experiment related to quantum erasure. After plotting the correlation measurement with and without blocking one of the polarization from the SPDC source (say, V polarization), I do not know how to work further on the gating time from the rate of the random...
  8. Rikudo

    Choose a ball at random from a randomly selected box - alternative way

    https://www.physicsforums.com/threads/choosing-a-ball-at-random-from-a-randomly-selected-box.1034377/ First of all, I would like to point out that this is the same exact question from what is being discussed in the thread above. In that thread, the problem is solved by adding the probability...
  9. A

    Probability involving Gaussian random sequences

    How do I approach the following problem while only knowing the PSD of a Gaussian random sequence (i.e. I don't know the exact distribution of $V_k$)? Or am I missing something obvious? Problem statement: Thoughts: I know with the PSD given, the autocorrelation function are delta functions due...
  10. Euge

    POTW Convergence of Random Variables in L1

    Let ##\{X_n\}## be a sequence of integrable, real random variables on a probability space ##(\Omega, \mathscr{F}, \mathbb{P})## that converges in probability to an integrable random variable ##X## on ##\Omega##. Suppose ##\mathbb{E}(\sqrt{1 + X_n^2}) \to \mathbb{E}(\sqrt{1 + X^2})## as ##n\to...
  11. A

    Characterizing random processes

    Hello. I would like to kindly request some help with a multi-part problem on identifying random processes as an intro topic from my stats course. I’m fairly uncertain with this topic so I suspect my attempt is mostly incorrect, especially when specifying the parameters, and I would be grateful...
  12. A

    Help with random variable linear estimation

    Hi all, I have a problem on linear estimation that I would like help on. This is related to Wiener filtering. Problem: I attempted part (a), but not too sure on the answer. As for unconstrained case in part (b), I don't know how to find the autocorrelation function, I applied the definition...
  13. A

    MSE estimation with random variables

    Hello all, I would appreciate any guidance to the following problem. I have started on parts (a) and (b), but need some help solving for the coefficients. Would I simply take the expressions involving the coefficients, take the derivative and set it equal to 0 and solve? I believe I also need...
  14. Lotto

    B How to calculate a random measurement error?

    I have seen this formula $$\sigma=\sqrt{\frac {\sum_{i=1}^{N}{(X_i- \bar{X})^2}}{N(N-1)}}$$ but also this formula $$\sigma =\frac{\sum_{i=1}^{N}{|X_i- \bar {X}|}}{N}.$$ Which of them is correct?
  15. C

    I Randomly Stopped Sums vs the sum of I.I.D. Random Variables

    I've came across the two following theorems in my studies of Probability Generating Functions: Theorem 1: Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then, ##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)## Theorem 2: Let ##X_1, X_2, ...## be a sequence of...
  16. A

    MSE estimation with random variables

    Hello all, I am wondering if my approach is correct for the following problem on MSE estimation/linear prediction on a zero-mean random variable. My final answer would be c1 = 1, c2 = 0, and c3 = 1. If my approach is incorrect, I certainly appreciate some guidance on the problem. Thank you...
  17. G

    A Diffraction from a set of concentric rings with random phase

    I have been considering the properties of a Diffractive Optical Element (DOE) consisting of a very large number of concentric rings of equal (small) width, where the thicknessses of the rings are such as to produce random phase shifts in the range 0 to 2pi. I think I understand the behaviour of...
  18. A

    Poisson random process problem

    Hello all, sorry for the large wall of text but I'm really trying to understanding a problem from a study guide. I am quite unsure on how to approach the following multi-part problem. Any help would be appreciated. Problem: Useful references I'm using to attempt the problem My attempt: For...
  19. A

    Sinusoidal sequences with random phases

    Hello all, I have a random sequences question and I am mostly struggling with the last part (e) with deriving the marginal pdf. Any help would be greatly appreciated. My attempt for the other parts a - d is also below, and it would nice if I can get the answers checked to ensure I'm...
  20. Fra

    A Peter Morgan (QM ~ random field, non-commutative lossy records?)

    "One way to ground everything in reality is to think purely about the records of experiments that are stored in computer memory. Very often, that's a list of times at which events happened." -- Peter Morgan, old thread meaning-of-wave-function-collapse "If we are to understand the relationship...
  21. A

    Break a Stick Example: Random Variables

    Hello, I would like to confirm my answers to the following random variables question. Would anyone be willing to provide feedback and see if I'm on the right track? Thank you in advance. My attempt:
  22. A

    Probability: pair of random variables

    Hello all, I would like to check my understanding and get some assistance with last part of the following question, please. For part (d), would I use f(x | y) = f(x, y) / f(y) ? Problem statement: My attempt at a solution, not too confident in my set-up for part (d). I drew a sketch of the...
  23. F

    I Random variable vs Random Process

    Hello, When flipping a fair coin 4 times, the two possible outcomes for each flip are either H or T with the same probability ##P(H)=P(T)=0.5##. Why are the 4 outcomes to be considered as the realizations of 4 different random variables and not as different realizations of the same random...
  24. D

    I Primes -- Probability that the sum of two random integers is Prime

    im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000. note that I am looking for a novel proof, not just some brute force calculation. (this isn't homework, I am just curious.)
  25. S

    Finding constant related to random variable

    Var (Y) = a2 . Var (X) (6.96)2 = a2 . (8.7)2 a = ± 0.8 But the answer key states that the value of a is only 0.8 Why a = -0.8 is rejected? Thanks
  26. M

    I Max of 3 random cards from deck vs max of 3 numbers from 1-13

    Hi PF! I am wondering the differences between the discrete and continuous case for expected value of minimum of 3 integers uniformly distributed from 1 to 13 vs 3 reals from 1 to 13. The real case is direct: ##F = ((x-1)/12)^3 \implies f = 3(x-1)/12)^2## for CDF ##F## and PDF ##f##. Thus the...
  27. M

    Probability that 𝑌>3𝑋 where 𝑋,𝑌 are 𝑁(0,1) random variables

    After plotting the above (not shown) I believe one way (the hard way) to solve this problem is to compute the following integral where ##f(x) = e^{-x^2/2}/\sqrt{2\pi}##: $$\frac{\int_0^\infty \int_{3X}^\infty f(X)f(Y)\, dydx + \int_{-\infty}^0 \int_0^\infty f(X)f(Y)\...
  28. M

    Random walks and destinations

    a) Let ##N_i## be the expected number of jumps to get to one of the square sides from minimal step number ##i## from the origin (so (1,1) would be ##i=2## since it takes 2 steps minimally to get there). Then we have: ##N_0 = 1+N_1## ##N_1 = 1 + 0.25N_0 + 0.5N_2 + 0.25## ##N_2 = 1 + 0.5N_1 +...
  29. M

    Given two random numbers in (0,1) what are odds that the quotient is closer to an odd integer?

    Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then $$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\ y < (2n + 1.5)x,\\ y > (2n + 0.5)x$$ for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...
  30. M

    I Given three random numbers between 0 and 1, how to evenly populate a sphere?

    Hi PF! Given three random numbers between 0 and 1, how to evenly populate a sphere of radius ##R## (assuming we use every point). I think it's similar to the 2D circle equivalent described here. Does this imply the PDF is ##4 x^2##, where the remaining analysis holds? Then one point is the...
  31. M

    A Magnetic sublevels of a random atom

    Hello! Assume I create an atom by some non-state-selective method (e.g. laser ablation, or hitting a proton on a target) and let's say that the atom is in a ##J=1## state. In the absence of magnetic fields, the ##m_J = 0, \pm 1## levels are degenerate. If I am to define arbitrary a z-axis (say...
  32. Steve Zissou

    I Distribution of Sum of Two Weird Random Variables....

    Hi there. Let's say I have the following relationship: x = a + b*z + c*y z is distributed normally y is distributed according to a different distribution, say exponential Is there a way to figure out what is the distribution of x? Thanks!
  33. D

    I Probability of Sum of 2 Random Ints Being Prime

    if I select two integers at random between 1 and 1,000, what is the probability that their sum will be prime?
  34. WMDhamnekar

    MHB Random digits appearance

    What is the probability that among k random digits, (a) 0 does not appear; (b) 1 does not appear; (c) neither 0 nor 1 appears; (d) at least one of the two digits 0 and 1 does not appear? Let A and B represents the events in (a) and (b). Express the other events in terms of A and B. My...
  35. WMDhamnekar

    Random digits appearance

    (a) The probability that 0 appears k times in k random digits is 0.1k So, It does not appear in k random digits is 1 - 0.1k. But author says 0.9 k. How is that? (b) My answer is same as in (a) that is 1-0.1k. Author's answer is 0.9k. (c)1 - 0.1k - 0.1k Author's answer is 0.8k. How...
  36. L

    Finding the distribution of random variables

    Hi. I have found the answer to a and c (I don't know whether it is correct) but I do not know what I should find in question b. Is my method correct and how should I solve part b? Thank you for your help!
  37. A

    A The normal equivalent for a discrete random variable

    De normal distribution has the following form: $$\displaystyle f \left(x \right) \, = \,\frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\nu \right)^{2}}{\tau ^{2}}}}{\tau ~\sqrt{\pi }}$$ and it's integral is equal to one: $$\displaystyle \int_{-\infty }^{\infty }\!1/2\,{\frac {...
  38. K

    I Exploring Continuous Approximation of 1D Random Walk Steps (Reif, pg 14)

    Reif,pg 14. ##n_1## is the number of steps to the right in a 1D random walk. ##N## are the total number of steps "When ##N## is large, the binomial probability distribution ##W\left(n_{1}\right)## ##W\left(n_{1}\right)=\frac{N !}{n_{1} !\left(N-n_{1}\right) !} p^{n_{1}} q^{N-n_{1}}## tends to...
  39. WMDhamnekar

    MHB Properties of sequence of random numbers and how to generate random numbers?

    Hi, Would any member of Math Help Board explain me the highlighted area in the following paragraphs? Generating Random Distributions Now the only missing thing in previous cases is how would one generate a Uniform random, Normal random distributions. We therefore look to cover algorithms to...
  40. K

    Using Poisson random variables to calculate this probability

    I calculated the mean which is 78.4 And the Standard deviation is 5.6 I thought the answer would be (90^(-78.4)/78.4!)*e^-90 But looking back having a decimal factorial doesn't make sense I have the numerical answers for c)= 0.019226 and d)=0.022750 but I my solution was wrong. Any help on...
  41. frost_zero

    Random Thoughts: A Journey Through the Mind

    The weird thoughts you have at random times
  42. wrobel

    A Lagrange-D’Alembert Principle and random ODE

    Here is my paper. A criticism and other comments are welcome. Abstract: The Lagrange-D'Alembert Principle is one of the fundamental tools of classical mechanics. We generalize this principle to mechanics-like ODE in Banach spaces. As an application we discuss geodesics in infinite dimensional...
  43. D

    B Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1?

    As we know those relations are true: if a/b = c, then a = b*c and b = a/c Therefore if 1/ ∞ = 0, ∞ * 0 should be equal to 1 and 1/0 = ∞
  44. docnet

    Calculate the joint CDF of two random variables

    $$f_{XY}=1$$ $$dzdy=2xdxdy⇒\frac{1}{2\sqrt{z}}dzdy=dxdy$$ $$f_{ZY}=\frac{1}{2\sqrt{z}}\quad \text{on some region S}$$ $$F_{ZY}=\int^y_{g}\int^x_{h}\frac{1}{2\sqrt{z}}dzdy\quad\text{for some}\quad g(x,y),h(x,y)$$ im learning how to find the region S using a change-of variables technique
  45. docnet

    Are X and Y dependent random variables?

    (a) the agrea of the triangleses is 1, so γ one. (b) I'm not sure how to prove. i feel like ##X## and ##Y## are dependent because ##E(Y|X=0)=\frac{1}{2}## and ##E(Y|X=1)=0## so ##Y## seems dependent on ##X##. ##f_X=1-x## for ##x>0## ane ##f_X=1+x## for ##x<0## so X seems independent on Y.
  46. M

    MHB Map of random variable

    Hey! :giggle: What does it mean to give the mapping for a random variable? Do we have to give the outcome space and the probability function? Does it hold that $X: ( \Omega, P)\mapsto \mathbb{R}$ ? :unsure:
  47. docnet

    Continuous joint random variable

    (a) $$\int_0^1\int_0^1x+cy^2 dxdy=\int_0^1 [\frac{x^2}{2}+cxy^2]_0^1dy= \int_0^1\frac{1}{2}+cy^2 dy=[\frac{y}{2}+\frac{cy^3}{3}]_0^1=\frac{1}{2}+\frac{c}{3}=1$$ $$\Rightarrow c=\frac{3}{2}$$ (b) The marginal pdf of X is $$f_X(a)=\int_0^1 f_{X,Y}(a,b)db=\int_0^1 x+\frac{3}{2}y^2...
  48. M

    MHB Multiple choice test : random variable

    Hey! 😊 A multiple choice test consists of 10 questions. For every question there are five possible answers, of which exactly one is correct. A test candidate answers all questions by chance. (a) Give a suitable random variable with value range and probability distribution in order to work on...
  49. M

    MHB Game : random variable for net profit

    Hey! 😊 You participate in the following game : You toss a fair coin until heads falls, but no more than three times. You have to pay $1$ euro for each throw. If your head falls, you win $3$ euros. The random variable $X$ describes your net profit (profit minus stake). Give the values that $X$...
  50. A

    B It works but why? (Matching experimental data to a random equation)

    Hey guys, I've about a week left to submit my final paper for my trade degree in transportation. The paper is about an analysis of potential implementation of an electric car for direct deliveries in my area where I live. In part of it, I try to analyze how many possible trips a car like...