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

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5
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178 views

expectation value for minimum distance between random variables

note: edited to clarify boundary issue Suppose $x_i$, $i=1\dots n$, are randomly drawn from a uniform circular distribution between 0 and 1 (using periodic boundaries). Let $d_i$ be the distance ...
4
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36 views

Random 3D points uniformly distributed on an ellipse shaped window of a sphere

I know how to generate random points uniformly distributed on the surface of a sphere: ...
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41 views

Entries of a Haar distributed unitary matrix

The eigenvector matrix of a Wishart matrix is Haar distributed and that implies that the eigenvectors are uniformly distributed on a sphere. I'm interested to know what is the distribution of ...
4
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50 views

Pairwise spacings of an ordered sequence of uniform random numbers

Given an ordered list of $m$ uniform random numbers in the range $1$ to $n$ $$a_{1} \le a_{2} \le \ldots \le a_{m}, \forall a_{i}: a_{i}\in [1;n] \cap \mathbb{N}$$ compute the pairwise spacings of ...
4
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85 views

Completeness of random walks in multiple dimensions?

I was reading Artificial Intelligence: Modern Approach (Norvig and Russell), and there was a footnote that really caught my attention. I apologize if the problem is more in the domain of CS than ...
4
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640 views

Random graph connectivity, and the existence of isolated vertices

Here $G_{n,p}$ represents the Erdős-Rényi random graph model, where the graph has order $n$ and each edge is added independently with probability $p$. I am faced with proving the following claim: ...
3
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31 views

What is the most important test for a uniform random number generator?

What is the most important test for a uniform random number generator ? Is there a single most important test or a set? I am a using some analytically arrived at answers to probability problems and ...
3
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37 views

Can I solve this probabilistic problem or do I need more data?

This problem happened to me almost 3 years ago and I just discovered this site by luck, so I want some help to know if this problem is solvable (if so, I would like to know how to solve it as well as ...
3
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53 views

Fractals vs. “neatness” / order

I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch ...
3
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177 views

Simulation and Stochastic Processes

Supposing we want to take a sample from the distribution $p(x)=cp^*(x)$ where $c$ is the normalization constant and $p^*(x)$ is given by $$p^*(x)=0.5\exp(-(x-\mu_1)^2)+0.5\exp(-(x-\mu_2)^2).$$ ...
3
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0answers
444 views

Notation for sampling random variate

Is there standard notation for sampling a value from a probability distribution? Like, if I had a random variable $X$, setting $x$ to whatever value I happened to sample from $X$ on this occasion? I ...
3
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804 views

Bays-Durham Shuffling

Reading "Random number generation and Monte Carlo methods" by James E. Gentle, I encountered an example concerning the Bays-Durham Shuffling algorithm whose result I can't reproduce. Basically, such ...
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21 views

What is the test of for randomness

How do you tell if a large sequential sample taken from a 0-1 rectangular parent - and presented as random - is truly random?
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58 views

Randomness in a sequence

For a function $f$ consider a random sequence $a_{n+1}$ can be either $a_n+f(a_n)$ or $a_n-f(a_n)$ Given that the next term in the sequence is subtracting $f(a_n)$ from the previous term 50% of the ...
2
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56 views

“Alice and Bob” problem with matrices

Alice has a binary matrix $A$, and Bob has a binary matrix $B$. Both matrices are of size $n \times n$. Alice and Bob want to check if the matrices are equal except for exactly $1$ entry. Alice has to ...
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38 views

Knuth shuffle : Is there a reciprocal to the factorial?

I have looked into the Knuth collection shuffle algorithm with pseudorandom number generators. They say that a PRNG with a seed state of $19937$ bits (like one of the Mersenne Twisters) can shuffle a ...
2
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0answers
62 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...
2
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62 views

Random pair generation?

Suppose there are 6000 people, there will be a combination of $$\binom{6000}{2}$$ ways for 2 people to be chosen out. Now the task is to randomly choose 5000 pairs of people in the total 6000 ...
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93 views

Using an elliptic curve to create pseudo random number

I recently started learning about encryption. I read about how elliptic curves can be used to create pseudo random numbers (and how the nsa might have abused this fact to create a backdoor in ...
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68 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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69 views

Packing a larger sphere with smaller spheres in high dimensions

We were discussing today the probability of leaving a point uncovered while trying to fill a larger sphere by randomly throwing in smaller spheres. Here's the argument: We are working in ...
2
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0answers
137 views

Finding functions where the increase over a random interval is Poisson distributed

I'm trying to construct a type of function $f(t_1, t_2)$ that counts the number of deterministically simulated Poisson events between two points in time. We can use a single valued function ...
2
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0answers
95 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
2
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118 views

Can every string of numbers be found in the number pi (cfr. infinite monkey theorem)?

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of ...
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0answers
76 views

What's the definition of a random number?

What sequence of numbers can I call as random number? What's the right way of getting $n$ random numbers? Are the numbers generated by "dice", too known as random numbers? Can a machine (computer), ...
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282 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 ...
2
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93 views

Approximability of continuous sampling by discrete sampling - definitions?

Are there standard definitions that express the notion that sampling from a given continuous random variable can be approximated "to any desired degree of accuracy" by sampling from an appropriately ...
2
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0answers
48 views

Generating random vectors in a n-ring

I've seen many different approaches to generate a random vector in the ($n-1$)-sphere and in the $n$-ball. One of them is generating a normal n-vector v (all components $x_i\sim N(0,1)$) and then ...
2
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231 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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0answers
73 views

Probability of ball ownership

$N+M$ people play a game of balls. Initially, N people hold N green balls (each person holds a ball), and M people hold no balls. Assume $M<N$. Then, M red balls are divided randomly - each ball ...
2
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48 views

Generate random numbers that are sums of hidden variables

I have $Y = A X$ where: $Y$ is an $m$ vector of random variables $X$ is an $n$ vector of random variables, uniformly distributed in $[0, 1]$ $m \ll n$, ($n$ ~ $m!$) $A$ is an $m \times n$ $0/1$ ...
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100 views

Modeling Sample Covariance Matrix based on concepts from Random Matrix Theory

I am working on a signal processing problem where I want to model the measurement sample covariance matrix (SCM) as random matrix and hence use the results from Random Matrix Theory (RMT). Let ...
2
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93 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
2
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124 views

LFSR with limited numbers of runs?

Is there a way to construct a linear feedback shift register (LFSR) which outputs no more than k consecutive 1s or 0s? (It would have to be not a maximal-length ...
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0answers
12 views

What does one-cut random matrix mean?

I am quite new to random matrix theory and recently I encountered the so-called "one-cut random matrix model" and even "two-cut" in physics. So what exactly does it mean?
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26 views

Understanding Sobol sequences

Can someone explain to me in simple terms, how Sobol sequences work? The wikipedia article is fairly technical. They look pretty interesting. So I shall describe (whatever little I know) in short the ...
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0answers
40 views

Why does a Gaussian process have a gradient whose determinant is Gaussian?

I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry. Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) ...
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26 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...
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21 views

Ideal shrinking of discrete randomness source

I have a discrete randomness source that emits random integer numbers in range [0..N) with uniform distribution. I need to reduce this distribution, limiting it to ...
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0answers
35 views

Calculate expected value E(|x-y|^2)

I have two random variables (X and Y) that are uniformly distributed from 2.16 to 6.81 both. And I need to find E(|x-y|^2). Is this correct: ...
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0answers
32 views

random nonsingular matrices using matlab

Does anybody know how to generate a random nonsingular matrices using matlab? I use sprand (m, n , dens, 1)function to specify the condition number to be 1 right now.But it is too slow.Is there any ...
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0answers
14 views

Raffle between different groups composed by different numbers

I've got this issue, I need to prepare a raffle between teams for a cars race. Cars are grouped by teams. Rounds are 1:1, composed by different manches until the cars are done. Total number of cars is ...
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30 views

Probability of a specified sequence in a random data set

This is a problem which I have encountered while programming, but I imagine this community would be better able to solve it. Suppose we have a number, N, of boxes in a row. In each of these boxes is ...
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82 views

Generate correlated random numbers precisely

Let's assume I want to generate k samples of n random numbers, that are correlated according to a given correlation matrix C (e.g. $n = 3$): ...
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0answers
45 views

determining the next random number pseudorandom number generator?

I have given 3 numbers let's say basic example x_0=5, x_1=6 and x_2=2 and modulus p is 7, ...
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0answers
21 views

generating locally random permutations

I have an intuitive notion of 'local randomness' that I am trying to make precise and understandable, and I am running into a bunch of problems. A quick web search failed to find anything relevant in ...
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0answers
46 views

Probability of random functions where domain equals co-domain

Given random function defined by $f: [n] \rightarrow [n]$, chosen uniformly, what is probability that the function is injective, surjective, or bijective? If $[n]$ is a set of discrete elements, ...
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96 views

Randomized packing of items

An adversary gives you a set of items whose total size is $x$ (he gets to choose how $x$ is distributed. e.g. there may be $k-1$ items of size $\frac{x}{k}$ and 2 items of size $\frac{x}{2k}$). The ...
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21 views

Determinate State of Linear Congruential Generator from Results

I am curious on how someone would go about determining the state of a Linear Congruential Generator given its output. X(n-1) = (aX(n) + c) mod p Since the values ...
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18 views

Good notation for many random points approximating an area.

I'm trying to say that as the number of random coordinate points points you plot approaches infinity, it is equivalent to an area integral where each point is an infinitesimally small $\mathrm{d}x ...