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

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2answers
38 views

Sum of Binomial Coefficient products

I am trying to prove that $$\sum\limits_{y=0}^d \frac{{2x \choose y} {2d-2x \choose d-y} }{2d \choose d} = x $$ So far, I have tried using induction on $d$ but I am having trouble using the ...
11
votes
2answers
744 views

Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
-1
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1answer
24 views

function of a random variable problem [on hold]

Let $X$ be a random variable uniformly distributed over $[a,b]$. Let $Y=(X-c)^2$ where $c$ is the constant such that $c \epsilon (a,(a+b)/2)$. Find the density of Y. Please help me.
4
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0answers
40 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: ...
0
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0answers
11 views

distance between random points in two non-overlapping circles [duplicate]

I have had asked a question yesterday, got a link of the book but can not find it anywhere to buy or loan. Can any one point me towards a formula or distance distribution for distance between two ...
11
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5answers
6k views

What's the difference between stochastic and random?

What's the difference between stochastic and random? I've read in the portuguese wikipedia that there's a difference, but I still didn't see this point on english wikipedia.
-1
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1answer
46 views

How can I use a Gamma Random Variable to Aproach the Expected value of a exponential random variable function?

I´m working on trying to approach the value of $E\left[ \dfrac{e^x}{x+1} \right]$. Where $x$ is an exponential random variable. All that data I have to work with is a gamma random variable with ...
0
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2answers
47 views

Solved - Use Random Number to Derive # based on Probability Table

Update I was able to derive the algorithm and implement it into excel. Thanks for the formula. Something like: ((z-xlbound)/d(x)*d(y))+ylbound See original sheet at end of post Original Post ...
3
votes
2answers
32 views

How to check if a sequence is random?

When I was thinking about various types of pseudo-randomness, the following question struck me: Suppose that a sequence $a_n \in \{0,1\}$ is given. Is there a way to check if it is genuinely ...
12
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1answer
288 views

Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from ...
2
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2answers
707 views

What does relaxing the iid assumptions mean? Intuitive and technical perspectives.

I believe the most restrictive assumption we can place on a series of observations is that they are iid. It is possible to relax these assumptions. For example relaxing the independent distribution ...
3
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0answers
63 views

Are humans capable of thinking of a series of random numbers? [closed]

I read in a book today ( the computer music tutorial by curtis roads), that humans are not capable of imagining any long series of truly random numbers. Apparently, the only way to generate a series ...
0
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1answer
29 views

Selecting a random orthogonal polygon

For a certain demo application, I want to create at random a rectilinear polygon with a given number of corners. Selecting random $x$ and $y$ coordinates of each corner is not a good method, since ...
2
votes
0answers
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?
3
votes
1answer
428 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 ...
1
vote
1answer
137 views

how to Evaluate integral of density of Wishart matrix

Let $X_1 \cdots X_N$ are $N$ number of $m$ Dimensional Independent Complex Gaussian Random vectors Such that: $$ X_j \sim \mathcal{N}(\mu,\Sigma)\; \forall \;j=1 \cdots N$$ Let ...
1
vote
2answers
38 views

Expected Value of Identically distributed random variables

I have a very quick question regarding the expected value of two random variables $X,Y$ that are identically distributed and not necessarly independent. Is this equation valid? $E[XY]=E[X^2]$ If ...
0
votes
0answers
15 views

Random variable variance

I have the model yi=β1+β2Xi+ui where ui∼iid N(0,σ2). I estimate β1 and β2 by drawing a straight line between the first (x1,y1) and last dot (xn,yn). So, β̂ 2 will be the slope of this straight line. ...
0
votes
0answers
25 views

The probability that uniformly distributed integers sum to a given integer

A recent CTF had a problem involving the summation of randomly distributed integers. Specifically: Consider a set $\{X_m\}$ of $M$ integers uniformly selected (with replacement) from the set of ...
1
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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?
0
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1answer
23 views

Translate the word phrase into a variable expression. [closed]

The number $a$ is increased by the number $b$. A. $a-b$ D. $a \div b$ C. $a+b$ D. $a \times b$
1
vote
2answers
19 views

Random number distribution from a different distribution

Suppose I have a random number generator that generates random numbers $x$ with a normal distribution $p(x) \propto e^{-x^2}$ (modulo normalization, but lets keep it simple). Now, out of these ...
0
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0answers
9 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
0
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1answer
16 views

Generate random results in a continuous field

How can we generate random results for a field like economical predictions where there is no limited number of results (contrary to a coin with 2 results) and also contrary to a random walk with steps ...
123
votes
24answers
10k views

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
3
votes
3answers
277 views

Deterministic random numbers generator using $p^n \mod q$

I figured that I can create a deterministic "random" numbers generator by utilizing a bit of "magic" that I picked up from some cryptography. However I seem to have missed a detail. Basically the ...
0
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1answer
23 views

How Many Unique Character String Can Be Made From 62 Characters?

I'm working on a programming algorithm and need a little math help. I'm in 10th grade and I think the question I'm asking is actually a permutation and combination logic question. Okay, so I've 62 ...
2
votes
0answers
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 ...
7
votes
5answers
638 views

Is the product of uniformly distributed numbers, uniformly distributed too?

My question is simple, I think. If we took two random natural numbers $a$ and $b$ uniformly distributed in a specific range $[c,d]$, is $ab$ a uniformly distributed too? What if $a$ and $b$ are not ...
2
votes
4answers
150 views

How to efficiently generate a set uniformly distributed numbers that add to $n$.

I am in need of a more generalized solution to this problem. I have a random number generator that generates numbers from 0 to 1. Using this, I want to find $r$ numbers that add to $n$. How do I do ...
2
votes
2answers
46 views

How to find out number of possible outcomes by trying over and over?

While working on my network exploration tool project, I've ran across the problem of reliably determining number of possible exit addresses of a tunnel with single entrance. I've came up with ...
0
votes
0answers
11 views

What it means by “asymptotic normality” properties of a random matrix?

I know that for the case of a random variable and a random vector, one can using (multivariate) density of normal distribution and concepts of convergence to define an asymptotic normality of a random ...
1
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0answers
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 ...
0
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0answers
8 views

Distribution of cut-off pseudo random numbers via linear congruential generator

I am currently using congruential prng as described here: Wikipedia. Now I need pseudo random numbers in the interval $[0, N), n < m$ where $m$ is the divisor of the modulus calculation. I want to ...
0
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0answers
35 views

Is the probability of variable independence = 0?

I understand the concept of independence to be dichotomous- events are either independent or dependent. And while there are infinitely many ways for events to be dependent and only one way to be ...
1
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2answers
43 views

What is the expected value of this sum (drawing slips from a bag)?

Suppose there are thirteen slips in a bag, labelled 1-13. If I draw a 10, 11, 12, or 13 then I stop adding to the sum and return the sum. If not, I add the number I draw to the current sum and place ...
0
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0answers
36 views

Dimensional Consistency in Grids used in Optimization

I am working on an optimization problem in the research I am doing and my partner and I have found that in order to quickly converge on a solution using a specific PSO (the firefly algorithm - it's ...
31
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3answers
6k views

Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
1
vote
1answer
20 views

Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
2
votes
1answer
29 views

The probability of getting a certain image by random pixelation

Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb. Just suppose that I've got a photo/image with 320x240 resolution and 24 bit ...
5
votes
1answer
189 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
4
votes
2answers
510 views

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...
1
vote
1answer
30 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
2
votes
0answers
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 ...
5
votes
2answers
1k views

What is golden ratio doing in this computer code?

In this file (related to random number generation), there is following line: private const int MSEED = 161803398; which reminds on golden ratio. How come ...
12
votes
2answers
1k views

Apparent Paradox in the Idea of Random Numbers

This question is a bit less than rigorous, but it's only because I don't know how to formulate it rigorously. Suppose there was some machine, or function, or whatever that could output a random ...
18
votes
8answers
2k 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 ...
0
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0answers
16 views

Spherical Sampling of Projected Disk

Given a 2D solid disk centered at some 3D point $\vec{x}$ with radius $r$ and normal $\vec{N}$, I need to compute a random unit vector from the origin that hits the disk. The vector needs to be ...
0
votes
1answer
16 views

PRNG to generate a known set of values

Say I want the first N numbers generated by PRNG to be some list [n1, n2, n3, n1, nx...]. Is there any way to accomplish this (basically find the appropriate seed ...
3
votes
1answer
72 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...