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

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7 views

Covariance of a random variable and its mean estimator

I was wondering if there is a common solution for the covariance of a random variable $x[n]$ (i.i.d) and its mean estimator $\hat{\theta}$. Mathematically, \begin{equation} \sigma^2 \left ( ...
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1answer
64 views

Solution for the Tall movie-watchers problem!

The question which I write is a changed format of what I have faced, in my project. I changed it to an attractive question and named it by "tall movie-watchers problem": Suppose that a small cinema ...
3
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0answers
25 views

Magnitudes of roots of random polynomials

I'm looking at the roots of random polynomials with integer coefficients, and constant term=leading term = 1. Using the Mathematica code ...
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1answer
72 views

Dividing gems by random permutation

A group of people have found a treasure of gems: $G=90$ green and $B=990000$ blue. They decided to divide it among them. Since there are more people then gems, they decide to order themselves in a ...
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1answer
39 views

Unexpected behavior with random numbers

I am trying to write a program dealing with random number generation, but I figured this would be the best forum to ask this. My program will, given each of the March Madness teams' probabilities of ...
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1answer
30 views

Random Shots and percentages behind it.

Just to be clear I am rather bad at math, however I am making a webservice and need to understand the math behind it in order to recreate it in javascript. So here goes.I am trying calculate the ...
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0answers
39 views

Covariance and correlation of summations of independent random variables

This is the problem: There are 2n − 1 independent random variables X1, 𝑋2, ... , 𝑋2𝑛−1. The expectation E(Xi) is μ for all i = 1,...,2n−1. The variance Var(Xi) is σ2 for all i = 1,...,2n−1. Let ...
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0answers
38 views

Question about noise term in SDEs

Do any properties/assumptions of SDEs prevent the noise term from being extremely large? Using a simple population growth model as an example, $\frac{dNt}{dt} = (r_{t} + W_{t})Nt , N0$ given, ...
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2answers
107 views

Does randomness exist? [closed]

I've been plagued with this question for a few years now and wanted to know what others think. Does true randomness really exist? In mathematics, a random process is based on the concept of random ...
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0answers
15 views

Generate list of random items without dublicates

I need to generate list of random int items without duplicates. for example: n = 6( 0, 5, 2, 3, 1, 4). I write simple algorithm based on ...
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0answers
27 views

Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots ...
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1answer
57 views

Are there any Martin-Löf random reals that are computable?

For example, Chaitin's constant is both Martin-Löf random and uncomputable. Are there any examples of numbers that are Martin-Löf random but computable?
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0answers
18 views

Fourier transform of a real white noise on a 3D cubic lattice

I'm facing the following problem: I have a cubic domain of side $L=2\pi$; this domain is divided in a cubic grid, each side is divided in $N$ points, where $N$ is an even integer number. the ...
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0answers
30 views

Sum of random numbers and e constant [duplicate]

I recently came across this interesting fact: Take some (pseudo)random numbers between 0 and 1. Now sum this and count how many you need in order for their sum to be greater than 1. If you repeat ...
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1answer
18 views

What is the average minimum distance between two Sobol points?

Having the first n points of a d-dimensional Sobol sequence, what is the average Euclidean distance from one arbitrarily point to its nearest neighbour?
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1answer
30 views

Covariance of random variable as a function of distribution of noise

Consider the following stochastic difference equation \begin{equation} x(t+1) = x(t) + \nu(t+1) \end{equation} where, $x(t)\in\mathrm{R}$ be one dimensional and $\nu(t)$ be the disturbance with an ...
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2answers
165 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
0
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0answers
22 views

Chi squared uniformity test for randomness

I'm evaluating some pseudorandom binary sequences, and I have a doubt. To begin with my reasoning was, let's apply a double tailed test to the chi squared statistic, because since I'm looking to see ...
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2answers
61 views

Probabilities seem to be growing exponentially

We have instituted random drug testing at our company. I was charged with writing the code to generate the weekly random list of employees. We've gotten some complaints because of people getting ...
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0answers
28 views

Method or Algorithm to produce near-accurate probability selection

Are there any known algorithms or methods to accurately identify, using random occurrences, the most probable next occurrence(s) as a pre-determined length of number set, by providing an already ...
8
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3answers
114 views

Uniformly Random Tuples

Consider a multiset of natural numbers. As an example take $$ M = \{1, 2, 2, 3, 3, 3\} $$ If we treat copies of the same number as indistinguishable, there are 8 distinct 2-tuples we can form from ...
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0answers
30 views

How to perform sampling using O(log n) random bits per sample irrespective of the values of pi?

I am not able to find a sampling for the following problem. "Consider a problem of using a source of unbiased random bits to generate samples from the set S = {0,1,...,n-1} such that the element 'i' ...
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1answer
14 views

hypergeometric distribution and random sampling

Is there any simple and fast algorithm (to be implemented in Javascript) to obtain a sample from the hypergeometric distribution? My needed sample size is very large (100,000,000).
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0answers
29 views

Expectation of a convolution between a WSS random process and an LTI system

Let $X(t)$ be a wide sense stationary random process―i.e., its expectation is a constant and its autocorrelaton function is a function only of time differences―and let $Y(t) = X(t) * h(t)$ where ...
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0answers
21 views

Design a statistical test for random number generator?

Design a statistical test for random number generators, based on the following result. Then apply the test to any generator you want and explain the results. A coin is flipped consecutively until ...
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3answers
873 views

How can one create random numbers with special correlations?

Is it possible to create uniformly distributed real pseudo random numbers $x_1,x_2$, and $y_1,y_2,y_3\in$ $[0,1]$, subject to the following constraints: $$x_1^2+x_2^2=1$$ $$y_1^2+y_2^2+y_3^2=1$$ I ...
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0answers
14 views

spectral norm of a sparse Gaussian matrix

Suppose $G$ is an $m \times n$ matrix such that each entry of $G$ is a standard normal variable. We know that the spectral norm of $G$ scales as $\sqrt(m) + \sqrt(n)$. Now, given a set of indices $S$ ...
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1answer
46 views

Given a list of integers between $0$ and $99$, create a function that will fit all the integers in the list.

Okay, so my friend LOVES to play the lottery. He makes bets often with bookies on the last $2$ digits, getting $90:1$ on his money, a losing bet. He looks up 'systems' on social media on how to ...
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1answer
25 views

Multiple rolls of a one hundreded sided dice

Let's side I have a one hundred sided dice and a friend picks 8 random numbers from the dice. I understand that if I roll the dice one time, he has an 8% chance of getting the roll correct. My ...
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1answer
26 views

is it possible to implement random(0,1,..,m) with finite calls to random(0,1)? [closed]

that is, is there a function $f$ that $Y=f(m,X_1,X_2,...,X_{n(m)})$ where $X_i\sim B(1,\frac{1}{2})$ and $Y\sim U\{0,m\}$? e.g. when $m=2^k-1$,$n(m)=k$ and ...
2
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1answer
19 views

Random allocation of groups of objects to agents

I have a poorly specified random allocation problem, which I need help in trying to tighten the definition and consider an effective algorithm. I have groups of objects, each group containing at ...
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0answers
15 views

K-Uniformity of Infinite Sequence

A book on random number generators refers to the subject of infinite-uniform infinite sequences as being "random." I was wondering if anyone could shed light on the definition of K-Uniform Infinite ...
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2answers
106 views

Probability that an individual cheated and knows a random sequence.

Bob writes down a sequence of coin flips on a piece of paper and hides it away. He uses a coin flip to determine if he uses a 1 or a 0. Alice tries to guess the random sequence and may have looked at ...
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2answers
26 views

Deriving probability densitys

How does one derive probability densitys involving fractions? For example, let $X^2$ and $Y^2$ be exponentially distributed random variables with parameter $\lambda = 1$. Determine the PDF for $Z = ...
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0answers
34 views

Deterministic seeded shuffle

How can I find an injective function $f$ so that mapping that function over each element of the ordered sequence $[1\cdots{n}]$, yields a deterministic shuffle (random permutation) that is "good" ...
0
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1answer
37 views

What's duplication probability of the next discrete random number?

If I already have n random numbers(between 0-100), what's the duplication probability when I random next number? This is a chart represent probability(0-100%) of this question when n is between ...
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4answers
83 views

What would be the maximum number of guesses for a random number between 1 and 9?

How many guesses could someone get wrong if they were trying to guess a number between 1 and 9, with the integer always changing with each attempt? For example: If a random number generator had ...
2
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1answer
67 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 to do ...
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1answer
20 views

Blum Blum Shub pseudorandom numbers - clarification on prerequisites

By reading on Wikipedia, they say the following (these is also supported by the following work: http://www.daimi.au.dk/~mg/mamian/random-bits.pdf) The seed x0 should be an integer that is co-prime to ...
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0answers
57 views

How can non-binary data be tested for distinguishability from random?

Given a large string of Base-N digits (in other words, not necessarily binary), is there software or an algorithm that can output whether the data is reasonably pseudorandom or reveals a pattern? In ...
16
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1answer
277 views

Flaw or not flaw in Excel's RNG?

I have a question about my understanding of an article of B.D. McCullough (2008) about Excel's implementation of the Wichmann-Hill random number generator (1982). First, a bit of context The ...
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0answers
38 views

Random Matrix Theory Noise

Hello and Merry (past) Christmas! I am new to random matrix theory, was reading an article about how to improve a correlation matrix (for portfolio optimization). And everywhere i see this "noise ...
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0answers
45 views

What is the chances of a duplicate in this equation [duplicate]

I'm not very good at math; However I have a scenario where I'm trying to find the chance of duplicate for randomly generated data. In a nuttshell I have a "bag" with 62 different items, lets say a ...
5
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1answer
67 views

Choosing primes uniformly at random

I'm interested in efficient methods of generating prime numbers in a given range [a, b] (or with a given number of bits/digits, etc.). By "efficient" I mean minimizing time, randomness, and perhaps ...
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1answer
67 views

Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
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1answer
55 views

Randomness of a linear congruential genarator in jumbling values of an array

I am working on a school project and it requires a simple pseudo-random number generating algorithm. I thought of using a linear congruential generator for this purpose. This came to my mind as it ...
0
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0answers
34 views

Probability distribution of the upper limit of an RNG given finite set of results

Suppose an unweighted random number generator outputs a random integer in the range $[0,u]$, and suppose I generate $n$ numbers and store the maximum number generated, $m$. Given only $m$ and $n$, can ...
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1answer
41 views

Can you select random entry from unknown number of entries?

Imagine you've got entries coming in, without knowing when they end (how many will follow from now on). You're supposed to pick random one and be fair. You can't save the entries that passed but you ...
1
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1answer
20 views

Application of Borel-Cantelli for sequence of two parameters

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ ...
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2answers
61 views

Expected Value Of a Random Variable X

Consider an experiment that is successful with probability $0.8$. We repeat this experiment (independently) until it is successful for the first time. The first $5$ times we do the experiment, we ...