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

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2answers
31 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 ...
3
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1answer
381 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 ...
0
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0answers
32 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
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1answer
18 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
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1answer
23 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
186 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
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2answers
494 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
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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
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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
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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
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2answers
989 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 ...
17
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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
14 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
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1answer
14 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
71 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 ...
1
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1answer
30 views

All combinations for a picking from a set of three colored balls

I can't find this exact question. The specific question I have is this: If you have a bag with one red, one blue, and one green ball, the chance of picking any particular one is 1/3. Simple enough. ...
2
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2answers
28 views

Select an element without uniform distribution with a uniform random without iterations

I have N elements (numbered from 0 to N-1) and I must choose one but without the same probability. For example, I need that the 0 must happen 50% of times, 1 with a 25%, 2 with a 12.5%, etc. I don't ...
1
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2answers
36 views

Exercise on Random Variables

I´m struggling with a random variable exercise of a book I´m reading. Anyone has an idea of how to approach this problem? Thanks in advance :)
0
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0answers
51 views

Probability Question involving Probability Mass Function/Random Variables

Problem: When a paging system transmits a message, the probability that it will be received correctly by the appropriate pager is p. To ensure that the message is correctly received at least once, the ...
0
votes
2answers
33 views

Calculate random integer inside a range of real numbers

$$F : \Bbb R \times \Bbb R \rightarrow \Bbb N $$ $$F(\text{minReal},\ \text{maxReal}) = \text{randomInt} \in \left[\text{minReal},\ \text{maxReal}\right] $$ Let $r \in [0, 1)$ be a random value. How ...
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0answers
23 views

Select machines to minimise latencies between them

I am working in an optimisation problem. I am still trying to model it and solve it. The problem is: There is a number of different types of virtual machines. Each type has different hourly cost ...
0
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2answers
39 views

Expected value: Product of integers in a range

What is the expected value of $f(x)^k = f(x) f(x) \cdots f(x)$ if $f(x)$ is a function, that returns a random integer between $0$ and $x - 1$ inclusive?
1
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1answer
35 views

Generate random numbers with a modified PERT distribution

I want to generate random numbers based on the modified PERT distribution. The modified PERT distribution is a special case of the beta distribution and is defined as: $$f_X(x) = ...
3
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0answers
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 ...
1
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1answer
21 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
2
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2answers
640 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 ...
0
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2answers
37 views

Probability Random Variable question Need Help Please

You have a set of ten light bulbs - the lifetime of each of them being given by an exponential RV with mean 1000 hrs. Find the probability that.... (a) at least 7 of the bulbs function for 1500 or ...
2
votes
1answer
24 views

Why is the formula for generating Van der Corput sequences called an Inverse Radical Function?

The Van de Corput sequence can be generated using the following formula: $\phi_b(n) = \sum_{i=1}^N { a_j \over b^{i-1}}.$ where this can be defined as the "one-dimensional sequence defined by the ...
14
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3answers
1k views

choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries?

choose a random number between 0 and 1 and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds 1. What's the expected ...
0
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0answers
19 views

mean square convergence vs almost sure convergence

I saw a few examples that show that almost sure convergence doesn't imply convergence in mean square. Can anyone find an example of a random series that converges in mean square but doesn't converge ...
0
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0answers
34 views

Proving properties of Random Graphs

I am asking the question on a slightly abstract level and it may depend on the specifics but it would be great to have related references or ideas. Consider the random graph model $G_{n,p}$ where its ...
3
votes
2answers
83 views

What is the probability of a specific sequence of 11 digits occurring in a random sequence of one billion digits?

This isn't homework, I'm actually (please don't ask me why) wondering how likely it is that any particular 11-digit telephone number will occur in the first billion digits of pi. My probability course ...
0
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0answers
13 views

How to normalise logarithm depending on the size of the range of possible values

I'm attempting to write a function which can be used to generate weighted random numbers between a set range, the size of which can arbitrarily grow and shrink, depending on some modifying value to ...
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0answers
39 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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0answers
37 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
12
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1answer
255 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 ...
-1
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0answers
15 views

Combing properties of Random Hashes and passing the dieharder test

I have two streams of numbers, range A are 64 bit numbers and may be largely sequential. In fact one of the issues is I don't know the full range and properties of these numbers. The other range (B) ...
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0answers
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 ...
0
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1answer
39 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
4
votes
2answers
212 views

“Randomize” output of a Linear Feedback Shift Register for the same taps?

I'm using a (Galois) LFSR to sample a large array, ensuring that each entry is only visited once. I simply skip past the entries that exceed the array length. With the same taps then the array entry ...
2
votes
0answers
53 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 ...
114
votes
23answers
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 ...
0
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1answer
31 views

Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $E[||x||_2]$, $x $~$ ...
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0answers
20 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 ...
14
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4answers
6k views

Picking random points in the volume of sphere with uniform probability

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of ...
3
votes
1answer
72 views

What constitutes randomness?

To begin with, I am not a mathematician, so the question might sound dumb :) Anyway, I have been playing 2048 and thinking on randomness and some things just do not seem very straight to me, when ...
2
votes
0answers
30 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
votes
1answer
69 views

Primitive polynomials in LFSRs

I need help proving the following theorem. I found it many books but on every single one it says that they omit the proof because it is in every good textbook. THM Let $c(x)$ be a connection ...
1
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2answers
133 views

How to find binomial pmf with probability = another random variable

Assume that $Q$ is a random variable with density proportional to $q$ for $0 < q < 1$. Given $Q = q$, $N$ has a binomial distribution with parameters $n$ and $q$. What is the probability mass ...