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.
I have a file consisting of a bunch of words, with one word in each line. For example, consider this webpage, or this one, but instead of being online, there is a .txt file is locally on my PC. My aim is to pick a word at random from this list that starts with a specific given letter. Say I want...
I do not understand the formulas (6.11) and (6.12) in volume 1 of the Feynman lectures on physics, the entire paragraph between equations (6.10) and (6.12) is generally not very clear. Please explain (preferably in simple language, I'm 13). Thanks!
$$\begin{align*}
E[(A+B)^2]&=E[A^2+2AB+B^2]\\
&=E[A^2]+2E[AB]+E[B^2]\\
&=2E[AB]+E[B^2].
\end{align*}$$
Can the terms ##2E[AB]## and ##E[B^2]## be simplified any more? Thanks, friends.
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).
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
"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...
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:
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...
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...
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.)
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...
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}##: $$\frac{\int_0^\infty \int_{3X}^\infty f(X)f(Y)\, dydx + \int_{-\infty}^0 \int_0^\infty f(X)f(Y)\...
a) Let ##N_i## be the expected number of jumps to get to one of the square sides from minimal step number ##i## from the origin (so (1,1) would be ##i=2## since it takes 2 steps minimally to get there). Then we have:
##N_0 = 1+N_1##
##N_1 = 1 + 0.25N_0 + 0.5N_2 + 0.25##
##N_2 = 1 + 0.5N_1 +...
Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then
$$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\
y < (2n + 1.5)x,\\
y > (2n + 0.5)x$$
for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...
Hi PF!
Given three random numbers between 0 and 1, how to evenly populate a sphere of radius ##R## (assuming we use every point). I think it's similar to the 2D circle equivalent described here. Does this imply the PDF is ##4 x^2##, where the remaining analysis holds? Then one point is the...
Hello! Assume I create an atom by some non-state-selective method (e.g. laser ablation, or hitting a proton on a target) and let's say that the atom is in a ##J=1## state. In the absence of magnetic fields, the ##m_J = 0, \pm 1## levels are degenerate. If I am to define arbitrary a z-axis (say...
Hi there.
Let's say I have the following relationship:
x = a + b*z + c*y
z is distributed normally
y is distributed according to a different distribution, say exponential
Is there a way to figure out what is the distribution of x?
Thanks!
What is the probability that among k random digits,
(a) 0 does not appear;
(b) 1 does not appear;
(c) neither 0 nor 1 appears;
(d) at least one of the two digits 0 and 1 does not appear?
Let A and B represents the events in (a) and (b). Express the other events in terms of A and B.
My...
(a) The probability that 0 appears k times in k random digits is 0.1k So, It does not appear in k random digits is 1 - 0.1k. But author says 0.9 k.
How is that?
(b) My answer is same as in (a) that is 1-0.1k. Author's answer is 0.9k.
(c)1 - 0.1k - 0.1k Author's answer is 0.8k. How...
Hi. I have found the answer to a and c (I don't know whether it is correct) but I do not know what I should find in question b.
Is my method correct and how should I solve part b?
Thank you for your help!
De normal distribution has the following form:
$$\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 {...
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...
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...
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...
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...
$$f_{XY}=1$$
$$dzdy=2xdxdy⇒\frac{1}{2\sqrt{z}}dzdy=dxdy$$
$$f_{ZY}=\frac{1}{2\sqrt{z}}\quad \text{on some region S}$$
$$F_{ZY}=\int^y_{g}\int^x_{h}\frac{1}{2\sqrt{z}}dzdy\quad\text{for some}\quad g(x,y),h(x,y)$$
im learning how to find the region S using a change-of variables technique
(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.
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:
(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...