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

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0answers
87 views

Bayesian random walk

Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. ...
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3answers
2k views

Normalizing a Gaussian Distribution

Assuming a Gaussian distribution with mean of zero and standard deviation of one, I would like to normalize this for an arbitrary mean and standard deviation. I know you're supposed to add the mean ...
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1answer
234 views

Normal distributed rotation matrix in 3D

How can I compute normally distributed 3D rotation matrices with Mathematica? For 2D matrices I would sample a normal distributed angle and directly create a rotation matrix with: ...
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1answer
114 views

generating a random periodic function with bounded amplitude and bounded fourier coefficients

I would like to generate (i.e. repeatedly compute via a computer) a random periodic function $f(x)$ with period $T$ such that $|f(x)| \leq M$ and the kth Fourier coefficient $|A_k| \leq g(k)$ for a ...
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0answers
34 views

Slicing Up Uniform Random Rotation Quaternions

I'm generating uniform random rotations using quaternions. I am using the method attributed to Shoemake, which is discussed in another post (Uniform Random Quaternion In a restricted angle range): ...
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1answer
40 views

Would this method yield a truly random sequence?

A creates some content A, say a poem. He assigns an arbitrary number value to each letter of the alphabet. He then selects a random piece of content B somewhere on the web without analyzing it. The ...
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0answers
188 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 ...
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1answer
97 views

Probability of Specific event occuring between 2 events?

Forgive me beforehand for what may be a question with an obvious seolution, but I havent had statistics courses in quite some time. I have an Excel File of approximately 3000 Events, each event has a ...
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1answer
65 views

A philosophical question on randomness

I have read in some book the following "philosophical" statement : "Introducing randomness we can make unstable things stable". Is there any practical example of this statement.
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0answers
44 views

generating random number with a watch

I don't know if this is the right place to post this question. If not I will move the question with no problem. Does exists an algorithm to generate uniform distributed random numbers using a watch? I ...
3
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1answer
110 views

Probability sequential terms of a linear congruential generator are coprime

I was following section 3.1.2 of the Structure and Interpretation of Computer Programming where they calculate $\pi$ using Cesàro's theorem that the probability two randomly chosen numbers are coprime ...
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1answer
56 views

Expected length of the “greedy” increasing sub-sequence?

Given a sequence of random unique integers of length $n$, if I select every element that is the largest so far how how many elements should I expect to select? This seems superficially similar to ...
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1answer
105 views

Yahtzee Bar Game

A bar near where I work has a game where you pay $5$ dollars which gets you two chances of rolling $5$ dice and if roll results in all of the dice having the same number you win the running pot, ...
3
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1answer
182 views

Number of samples to predict the next number in a pseudorandom number generator

Let: $$R_{n+1} = (mR_n + b) \bmod{a} $$ Assume we know the values of $R_1, R_2, \ldots, R_L $. What is the minimum value of $L$ (if it exists) such that we can determine $R_0, m, b$ and $a$?
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2answers
152 views

Random number generation in $3D$

I have problem regarding random number generation. Suppose I have disc of radius $r$. $$ \begin{align} x&=r\cos(\theta)\\ y&=r\sin(\theta)\\ z&=0 \end{align} $$ I rotate the co-ordinate. ...
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1answer
111 views

Gaussian distribution random noise error

Why do we in general consider the errors/noise in measured data are distributed in the Gaussian form? What is its advantage over the Laplace distribution? Moreover, if we have to add some random ...
4
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2answers
136 views

Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere ...
2
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2answers
55 views

Manipulating random decimals

I've been slowly working my way into the world of AI and its representations of random. Well as you can guess this occurs from a call to random() resulting in a ...
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4answers
923 views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
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1answer
131 views

A consequence of the law of large numbers

Let $(X_k)_{k=1}$ be Poisson random variables with expectation $\mu$, let $Y_n = \sum_{k=1}^{n} X_k$. The weak law of large numbers states that, $$ \forall \delta>0, \forall \epsilon>0 \, \, ...
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1answer
56 views

Sum of poisson random variables on a lattice

Consider a lattice $\mathbb{Z_+}$ and immagine that on each site $i \in \mathbb{Z}_+$ there is a number of particles $X_i$, where $X_i$ are i.i.d. Poisson random variables having expectation $\mu$. ...
2
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1answer
74 views

law of large number modified statement

The weak law of large number states that, given $Y_n = \sum_{k=1}^{n} X_k$, where $X_k$ are random variables independent and identically distributed with finite expectation $\mu$, $$ \forall ...
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1answer
59 views

Distribution of the Inverse of a Random Variable

I am trying to figure out how to find the distribution of the inverse of a random variable. Say, $Y=X^{-1}$ where X can take negative values. The two ways I know to find the distribution of a random ...
2
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0answers
110 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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2answers
63 views

Pseudorandom numbers - Get a sequence of number not close each others

I have this function inside a software: a+(SeededRand(Round((SongTime-0.5+(Round(1000*c)))*1))-0.5)*b it creates a pseudo-random numbers sequence inside the ...
5
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1answer
140 views

Generate random numbers between a range such that no number comes twice.

Sorry if my question is stupid, math has been always a wild beast for me. I am an application developer. In one application I have a module which assigns a random 6-8 digit number and a serial number ...
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1answer
43 views

How to generate sequence like this?

Can you tell what algorithm can generate sequence $x_1, x_2, x_3, x_4, ...$ satisfying: $x_n$ is real, and always $0<x_n<1$. Every change between $x_n$ and $x_{n+1}$, such as increase or ...
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2answers
194 views

Can random number generators be used in compression?

This is basically a recurring thought I have from time to time and I suspect it is flawed, so my question is basically why can't you do the following: Basically, I imagine that given a seed and a ...
0
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1answer
74 views

Chernoff type Sum of independent random variables having exponential tails

Say I have n independent variables $\{X_1,X_2 \dots X_n\}$ with Expectation 0 such that $Pr(|X_n| > \alpha) < e^{-\lambda \alpha}$. Can we produce chernoff type inequalities for the sum of these ...
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1answer
49 views

Average number of $U[0,1]$ random numbers to reach $x$

Let $P(x)$ be a random process where you keep selecting random numbers, uniformly distributed between 0 and 1, until the sum reaches $x$. From memory, the expected value of the number of terms needed ...
2
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0answers
74 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), ...
0
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0answers
46 views

Property of a random distribution.

I have to get an integral in a previous post Help on an integral.. Some guru gave me hints of how to approximate its value, but I need approximation with largely varying $a$ and $b$. I realized that ...
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3answers
115 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 ...
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1answer
82 views

When is an infinite sequence of integers purely deterministic with no randomness involved?

I see in literature very different descriptions of what is a deterministic system such as: "... a system in which no randomness is involved in the development of future states of the system...>>>" I ...
12
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4answers
538 views

How to efficiently generate five numbers that add to one?

I have access to a random number generator that generates numbers from 0 to 1. Using this, I want to find five random numbers that add up to 1. How can I do this using the smallest number of steps ...
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1answer
26 views

Concentration of measure for sum of products

Let $X_i$, $i \in \{1 \ldots n \}$ be independent random variables taking value +1 or -1. I know that there exists a lot of results which talk about the concentration of $S_n = \sum\limits_i X_i$. Can ...
11
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2answers
504 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 ...
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1answer
96 views

Probability of higher occurrence of an element within a random permutation with repetition

I generate $n$ random numbers, each one from a set $X={1,\ldots,N}$, where $n\geq N$. This results in a random permutation with repetition of length $n$ over $X$. Ideally for me, each number of $X$ ...
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0answers
29 views

Halton - Sequence question

I'm writing a program where I need to take samples from a 2D array of pixels on a screen. I've implemented a halton sequence implementation in my code that can generate low discrepancy samples on the ...
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1answer
49 views

Generate a Normal in 3D Without Branching?

I have a vector $v$ in arbitrary 3D space ending at point B. In order to generate the next point -- C, I uniformly pick an ...
3
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1answer
113 views

What does a north-east pointing arrow symbol (↗) stand for in mathematics?

Is there a canonical definition of "↗"? I've seen it in a formulation of the Monotone convergence theorem as follows: Let X and Xn be random variables for which Xn ↗ X ... from which it would ...
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0answers
69 views

How to generate a random number including mathematical dispersion?

I need to generate not completely random number that would be influenced by functional dependency. Current solution I was looking was to give an algorithm value of statistical dispersion or spread ...
2
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1answer
117 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
78 views

How to generate a N*D random matrix with columns of unit length?

Is it possible to generate a N*D random matrix with columns of unit length? If not, I also think it is possible of generating a N*D random matrix and, after that, normalizing it in order to have ...
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1answer
91 views

What is the meaning of 'columns have unit lengths'

What is the meaning of this? In random projection, the original d-dimensional data is projected to a k-dimensional (k << d) subspace through the origin, using a random k × d matrix R ...
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2answers
81 views

What exactly does this physically mean?

Let X(w) be a real random variable on ($\Omega$ , P). The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$. For any set $ B \subset \Omega^{X}$ the probability of the ...
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0answers
219 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
97 views

How can I use the Box-Mueller method to generate generic normal distributions?

Using the box-mueller method, we can generate standar normal distributions from two uniformly distributed random variables. However, how do we use this method to generate normal distributions with ...
2
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1answer
140 views

Pick the highest of two (or $n$) independent uniformly distributed random numbers - average value?

With "random number" I mean an independent uniformly distributed random number. One Picking one random number is easy: When I pick a random number from $x$ to $y$, the average value is $(x+y)/2$. ...
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0answers
130 views

Transformation of pdf to calculate cdf

A random variable $X$ has a pdf given by $$\displaystyle f(x) = \frac{2(x+2)}{25}, -2 < x <3$$ Let $Y=X^2$ be a transformation of $X$. $1$. Is this transformation one-to-one? Find the ...