Questions relating to (pseudo)randomness, random oracles, and stochastic processes.

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Permutations in an Infinite List of Random Numbers

In an infinite list of random numbers from a to b, prove that in this list, there are all possible permutations of n numbers from the list, where n can be any number. Here are some versions of the ...
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1answer
1k views

Mapping random points on a sphere onto a uniform grid

Say I had an arbitrary sphere that is covered in a uniform triangle mesh of N elements with each element having a unique sequential index. If given the coordinates of a random point on the surface of ...
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0answers
19 views

scrambled sobol

I need to do a Monte Carlo simulation in high dimension (up to 1000) where using plain Sobol (with Kuo's direction vectors) as a random number generator is not good enough. Therefore I am ...
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1answer
32 views

Test to determine whether a huge series of integers is random or there is a pattern?

I have a series of integers in which some integers occur more than others. I want to know if there is a test to determine whether these integers are randomly distributed or some integers occur more ...
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1answer
540 views

Fast generation of Pareto-distributed randoms.

I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
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1answer
159 views

How many random samples needed to pick all elements of set? [duplicate]

If repeatedly picking a random element from a set, what is the expected number of times I'd have to pick before seeing all the elements of the set? Edit: when picking an element, it is simply counted ...
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1answer
405 views

Measuring degrees of randomness

Imagine, for simplicity's sake, that we have a set of numbers, each equal to either 0 or 1. Let's call each a bit. Rationally, if the set is completely random, and reasonably large, the probability ...
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2answers
46 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
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1answer
473 views

Probability distribution of the product of two independent complex gaussian random variables

I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that $\...
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1answer
40 views

How large must be the sample to determine the probability of a binary outcome?

Suppose a computer program randomly generates a "yes" or "no" answer upon request. You however do not know whether the probability of getting either answer is evenly split or following a different ...
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1answer
45 views

How to simulate a delta-correlated random process

I'm trying to do the simulation described in the paper attached, but there is something I don't understand. The author says that the random variables which satisfy the relation (Eq. (4) in the paper) ...
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1answer
43 views

Does this set of infinite binary sequences have positive probability?

The AMM article "What is a random sequence?" argues (at the end of Sec. 2) that if, from the set of all binary sequences, we remove those (countably many) that have "computable regularities", then the ...
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1answer
35 views

Number of random guesses needed to guess a number in a given set [duplicate]

I am not very good with probability theory and related stuff, so I would very much appreciate your help regarding a problem I have. Imagine arbitrarily picking an integer $n\in [1,N]$ I want to ...
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167 views

How to attack universal hash function based on finite-field arithmetic?

As per the Recursive n-gram hashing is pairwise independent, at best paper, I want to use the algorithm described in chapter 6 and 7 (page 7 - 10). The hash works as follows: Define a random ...
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0answers
352 views

Are these numbers “random”?

The figure below shows $2000$ points in $(x,y)$ coordinates that are supposed to be high quality pseudorandom numbers. However, when I zoom in on any area lots of points are lined up along line ...
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2answers
54 views

Are these numbers random? [closed]

Are these numbers random? If not, please give 2 reasons. $$0.001\ \ \ \ 0.002\ \ \ \ 0.003\ \ \ \ 0.004\ \ \ \ 0.005\ \ \ \ 0.006\ \ \ \ 0.007\ \ \ \ 0.008\ \ \ \ 0.009\ \ \ \ 0.01 $$
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0answers
19 views

Generating Failure rate function using Octave.

I'm not familiarized with programming using mathematics. My problem is described below. The lifetime T of a device has pdf Find the failure rate function. I have learned that Failure rate ...
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1answer
53 views

Logic behind Metropolis algorithm

I am using Metropolis algorithm to make a program for Ising model in Statistical Physics. In Ising model, we take a collection of spins with initial energy, say $E_i$, then we randomly flip one of the ...
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1answer
63 views

What is the average distance between two random points inside a circle?

Assume you have a circle with some radius r. What is the average distance between two random points inside the circle? (Edit: This is different from this already answered question, because here the ...
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5answers
74 views

Using the digits of $\pi$ to generate random numbers.

Let's say I've been captured by Russian operatives and am locked in a room with only one object: a book listing the digits of $\pi$. I'm told to generate a sequence of binary digits. If this sequence ...
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1answer
48 views

Terminology: is it random?

The topic of research of my master thesis is the use of probabilistic methods and models in music composition, particularly in the field of algorithmic music. As often is the case, artists tend to be ...
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0answers
60 views

Having rand2() function build rand5()

I was asked this question long time ago. Having a function $rand2()$ (in any computer language, "rand" means random) which returns $0$ or $1$ (two values only) with a uniform distribution, i.e. $$P(...
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4answers
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Is there a “most random” state in Rubik's cube?

Is there a state in Rubik's cube which can be considered to have the highest degree of randomness (maximum entropy?) asssuming that the solved Rubik's cube has the lowest?
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1answer
419 views

Compressing random numbers

I've been thinking about ways to compress the output from a (supposedly) random number generator. Let's assume for a moment that my computer can produce high-quality random numbers. I'm certainly not ...
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23 views

random binary sequence [duplicate]

Suppose we generate a sequence containing only 0's and 1's. We generate this binary string by randomly adding a 0 or 1 one at the time. We stop when the sequence contains k more 1's then 0's or k more ...
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0answers
51 views

Random walk visiting $k$ distinct points

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...
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0answers
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discrete random variable with uniformely distributed random variable

I hope you can help me because I have no clue where to start: Let X be a discrete random variable with $ p_k=P_X[X=x_k]=p(x_k) $for all $1\le k\le N$ for $N\in \Bbb N$ and distribuition function: $$...
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2answers
66 views

Random and non-computable numbers

Let $\alpha \in (0,1): \quad \alpha=0.a_1a_2\cdots a_n \cdots \quad$ where the $a_n$ are numbers generated by a physical generator of genuinely random numbers (if it exists). Than it seems that $\...
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1answer
20 views

Probability of maximum of a random walk?

Let us consider a random walk denoted by Sn and let Mn be the maximums of the random walk. Now let us consider that this random walk will end at some point k. SO I am stuck how to prove this equality: ...
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0answers
45 views

How to punish a riffle shuffle?

It is common knowledge that for a deck to be considered randomly sorted, at least seven riffle shuffles should be used. However, in my experience, very few people will take the time to complete seven ...
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1answer
250 views

What is a thorough method to manually generate a random number between $1$ and $100$?

The other day, I got an idea. I would like to generate a random number between $1$ and $100$ , however by hand. And only using simple tools like a desk clock and pen and paper. What might be a way ...
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0answers
23 views

Probability of the n-th dice matching in 2 groups of sorted random numbers

Say we have 2 groups of 6 6-sided dice. Each group of dice is rolled and then sorted so we have 2 groups of sorted numbers. What is the probability of each die in the group matching the corresponding ...
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1answer
32 views

Forced probability using a random number generator, mathematically correct or not?

What might happen if when using a random number generator, if a particular run is not so random, you "help it out" a little bit? For example, if we use a random number generator to simulate $10,000$ ...
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Is a “deterministic” subset of a random subset random?

Let $S$ be some set and consider $X \subseteq S$ of size $|X|=x$ u.a.r. (among all the subsets having this size). Now, use some properties of this set $X$ to find some subset $Y\subseteq X$ of some (...
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20 views

Arbitrary vs. random subsets: computing probabilities

Let $G=([n],E)$ be a graph having minimum degree $\delta(G) \geq (1-\delta) n$. For some $q=q(n)$, let $G_q=([n], E_q)$ be the random subgraph of $G$ obtained by deleting each edge independently with ...
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1answer
96 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 \end{...
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3answers
91 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, $...
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1answer
26 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
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1answer
49 views

Is there a statistical measure of bitwise entropy?

(Somewhat inspired by this website, particularly Section III. Also, I might be using a different definition of entropy than usual; what I am using is closest to the physics definition (the one I ...
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1answer
10 views

Extending Random Number Ranges

I am provided with a random number $\xi \in [0,1]$. I check if a particular $\xi_i \lt x$ is true and if so, I need to convert those random numbers within the range $[0,x)$ into uniform range in $[0,1]...
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Particle in a box

Say I have a point particle located at the center of a box and imagine that I give it a velocity v in some direction. It will bounce back and forth in different directions maintaining the same speed |...
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49 views

Sequence of non-independent coin tosses

Suppose that a sequence of coin tosses is due to be performed. Let $p_i$ denote the probability that the $i$th coin toss lands on Heads and let $X_i$ denote the corresponding indicator random variable ...
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2answers
5k views

Calculating license plate combinations

If there are $26$ available capital letters and $6$ available numbers $0$ to $9$ to complete a $6$ "digit" combination (a license plate for example) what is the formula for calculating the number of ...
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1answer
26 views

Invertible Uniform “PseudoRandom” Function

Perhaps this is better suited to a cryptography stack exchange, but I thought I'd try in mathematics in case this question is more obvious than I initially thought. I'm looking for a function $~f:\{1,...
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1answer
15 views

Why does the average of a set of random numbers to the nth power approach 1/(n+1)?

I got bored and started running a Java program to mess with stuff like this. I did a boatload of trials and averaged them all together, first for a random number squared. Quick pseudo-code: ...
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1answer
30 views

Extinction time of contact process on finite lattice

Suppose we have a contact process on a finite lattice. I'm asked to give a heuristic argument for the fact that the extinction time for the contact process is exponential in the size of the lattice ...
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8answers
3k views

3 random numbers to describe point on a sphere [duplicate]

I'm currently working on a problem involving computer graphics and got into a discussion about whatever or not constructing a 3d vector out of 3 random points uniformly distributed points between -1 ...
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0answers
45 views

Mathematical expression for randomly selecting from two integers

I wonder what is the shortest and correct mathematical expression for the below: $x =$ random$(1$ or $2)$ where there is equal probability of selecting 1 or 2. Thanks for your help.
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Limiting distribution of infinite sum of weighted bernoulli?

Let $p_n$ be some fixed pulse, for example $p_n =e^{-n^{2}}$ We have an infinite sum $y = \sum_{n=-\infty}^{\infty} a_n p_{-n}$ where $a_n$ are iid bernoulli random variables taking the values $+/- \...