Questions tagged [random]

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

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17 views

Transformation of Random Variables: How to relate these two approaches?

In the literature, I have found two approaches for the transformation of random variables $$p_{X}(x) =\int_{-\infty}^{\infty}p_{Y}(y)\delta\left(x-f(y)\right)~\mathrm{d}y$$ and $$p_{X}(x)=p_{Y}(f^{-1}(...
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9 views

Mapping outcomes of binary number generator to decimal range $[1, x] $with equal probabilities

I am looking for a function that maps each possible binary outcome from a binary number generator to a decimal range $[1, x]$ such that each value in the range has an equal chance of appearing. For ...
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1answer
14 views

Simple random Sample Without Replacement (SRSWOR)

Note that SRSWOR refers to Let U be a population of size N. From U, we first select a Simple Random Sample Without Replacement (SRSWOR), S_1 , of size n_1 . Then, from S_1 , we select a SRSWOR, S_2 , ...
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+50

Random domino tilings: Is this distribution well-defined, and how can it be sampled from?

I'd like to ask questions about a "random domino tiling of the plane". However, it's not quite obvious how to go about precisely specifying what this means. My first instinct was to do ...
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1answer
94 views
+50

Shuffle a poker deck between 4 players, with least required entropy

We are shuffling a standard poker deck of 52 cards between 4 players, each getting 13 cards. The order of cards for a particular player does not matter. A naive algorithm is to first shuffle the whole ...
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18 views

Find the probability that a function takes a particular value [closed]

Consider a function $f$ which takes a list of integers as argument and is defined like - $f(L) = 1 $ if $\exists$ x in L such that $x>=10$ $f(L) = 0 $ otherwise Now consider a list $L$ having $n$ ...
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2answers
29 views

Trajectory of stationary stochastic process not in $\mathrm{L}^2$

Let $(X(t))_{t\in T}$ with $T\subseteq \mathbb{R}^n$ be a (strictly) stationary stochastic process. In the book 'Fourier Analysis and Stochastic Processes' Pierre Brémaud states on p.157 (Chapter 3.4....
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42 views

How to choose $x$ according to a pdf that looks like $1-\exp(-x)$ when I can't evaluate the cdf

I have the pdf of a random variable $x$ and it looks like $1-\exp(-x)$. I want to find a way to construct a method that picks random $x$ according to this distribution. Let's say by using another ...
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1answer
20 views

How to add a non-zero mean to the equation of state of Kalman filter

The measurement data of the laser gyro is used to establish the noise random process model, and then the Kalman filter state equation is established through the model parameters. First, remove the ...
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1k views

Finding period of a recursive sequence defined by modular operator?

If $f(x)$ is defined as : $f(x)=i$ ,if $x\equiv i$ (mod n), $0\leq i< n$ How can I prove whether the following recursive sequences are periodic or not? $$x_{i+1}=f(k_0+k_1x_{i}+k_2x_{i-1}+...+...
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20 views

Existence of stochastic process with square-integrable realizations

Consider a stochastic process $X = (X(t))_{t\in\mathbb{R}^n}$ with $n\geq 1$ and each $X(t)$ is a real- or complex-valued random variable with \begin{equation*} \mathbb{E} ( X (t)) = 0 \quad \text{and}...
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32 views

Bounding the quadratic form of a random matrix

I encounter a problem of bounding the smallest eigenvalue of the following $p\times p$ matrix: $$ A=\sum_{j=1}^{n}Y_jY_j', $$ where $Y_j=n^{-1}\sum_{t=1}^{j}x_t+n^{-1/3}\psi$. Here $\{x_t\}$ is a ...
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1answer
24 views

Does a truly random sequence in the range of x..y average to the average of x and y?

I recently started wondering if the average of a truly random sequence between numbers x and y has to be the average of x and y itself. This little JavaScript function seems to prove it (uses ...
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1answer
32 views

Probability problem on Unif(0,1) random variables

Given three iid Unif(0,1) random variables a, b, and c, what is the probability that c is the maximum if a+b < 1? The probability distribution of a+b for a+b < 1 is a+b and for 1 < a+b < 2 ...
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74 views

Approximate way to find function of multiple random variables

I want to know if there are approximate solutions to the joint distribution of functions of random variables, where the variables can be only normal or lognormal. The functions would be using only ...
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1answer
70 views

Selecting a random number from $(0,1)\cup(1,\infty)$

A random number $a$ is selected from $(0,1) \cup (1,\infty)$. Let $A $ be the event that (a random number from $(0,1) \cup (1,\infty))$ $ \in (0,1)$. Let $P(A) = x$ Let $B$ be the event such that (a ...
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10 views

Sampling from Haar measure [duplicate]

Is there an easy algorithm that outputs a random element of $SO(3)$ distributed according to Haar measure on $SO(3)$? Or more generally, replace $SO(3)$ with any compact matrix Lie group. Note: I'm ...
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2answers
135 views

Pseudo random ordering of integers

I remember an old retro effect for a screen resolution of $320\times 240$. You would iterate the pixels in a linear fashion so there are $76800$ pixels. You could iterate then one by one starting at ...
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1answer
44 views

How is the sample space of a random variable defined?

I was watching this example where the professor said that if we have 2 i.i.d RVs, both of them being binomial random variables, then we can only add them up if their sample spaces are the same. I am ...
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23 views

Can information transmission be proven in a Rule 30 ECA?

(This is hopefully a clearer version of an earlier post of mine.) I have been spending lots of time on the open challenge of proving the aperiodicity of the central column of a rule 30 cellular ...
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1answer
43 views

Find $E[X]$, when $X$ is a random number from Uniform$(0,1/k^2)$

Question: We have a Random Number Generator defined as: $\displaystyle U_{K} \sim \operatorname{Uni}\left[ 0,\frac{1}{k^{2}}\right] ,\ k\geqslant 1$. We will take a random number from $I\sim \...
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27 views

Applying a Stratified Random Sampling Technique to Real-World Problems

I am currently working on a paper that involves applying a stratified random sampling technique to a real-life problem. Say, I have a population in which I have divided the population into 200 strata ...
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1answer
22 views

Computing random numbers from an arbitrary (non-Gaussian) multivariant CDF

As part of a test harness I need to create random pairs of numbers according to a specified 2D CDF (Cumulative Distribution Function). The CDFs change over time and are the result of some opaque (i.e....
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1answer
80 views

generate any random integer

I apologize in advance as I am not very experienced with any formal notion of randomness. The title says most of it: I want to generate a random integer within a reasonable time, where every integer ...
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3answers
92 views

Product of random numbers.

Find the constant $p$ such that the product of any (positive) number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to ...
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1answer
46 views

Probability that a 2-d random variable falls into an area equals the double integral of its PDF?

Let $x:\Omega\to\mathbb{R}^2$ be a 2-dimensional r.v., and $f(p,q)$ be its probability density function. For what kind of $D\subseteq\mathbb{R}^2$, $\mathrm{Pr}[x\in D]=\iint_D f(p,q)\mathrm{d}p\...
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3answers
2k 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 ...
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2answers
40 views

Do all LCG-based PRNGs suffer from predictable patterns?

I needed to produce trivial (low-quality) random integers and remembered how simple linear congruential generators were to implement from school: Went to Wikipedia, found the first example which ...
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33 views

Probability question: 2 people meet from other sides of the world, who find they have lived in the same 3 property numbers in the same order

To me the probability of this happening seems almost impossible... I'm British and a friend of mine is Chinese. I showed her a letter I had received which had my address on it. She spots my apartment ...
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1answer
27 views

In a random walk starting at A and then moving to any adjacent point with equal probability. Find P(reach B before E)

It seems to me that for symmetric reasons the answer should be $\frac{1}{2}$. I tried to prove this by separating paths that did not include A and those that did. The probability of the former is $\...
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1answer
31 views

Simultaneous diagonalisation of matrices by diagonalising a random linear combination of these matrices

According to a reply on Mathematica SE numerically finding a matrix that simultaneously diagonalizes a set of pairwise commuting diagonalizable finite complex matrices $\{M_1,\ldots,M_n\}$ can be done ...
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1answer
25 views

$X_t$ and $Y_t$ are id for each $t$. $T_1$, $T_2$ are iid. Are $X_{T_1}$, $Y_{T_2}$ id?

I am working on a problem and as an intermediary step I think I need to use the following: $X_t$ and $Y_t$ are identically distributed for each $t\in[0, \infty)$. $T_1$, $T_2$ are iid, continuous RVs. ...
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0answers
17 views

Efficient way to sample random element $y \in \mathbb{Z}_{mn}^*$ such that $ x = y \mod n$ where $x \in \mathbb{Z}_{n}^*$

Efficient way to sample random element $y \in \mathbb{Z}_{mn}^*$ such that $ x = y \mod n$ where $x \in \mathbb{Z}_{n}^*$ Let $x \in \mathbb{Z}_{n}^*$. I want to find an efficient way to sample a ...
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1answer
26 views

NP problems understanding on specific example

Let's say we have a game where we should discover a number of length n, generated by RNG. If someone says us a solution it is easy to verify - we enter the number into machine and it compares it to ...
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1answer
830 views

Can we draw the graph of the derivative/integral of a function by using the graph of the function only?

Consider a function say $F(x) = x^2 + 5\sin x$ then we have it's derivative as $F'(x) = 2x + 5\cos x$ and thus we have the graph of $F'(x)$ quiet easily but can we plot a graph using only the graph of ...
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2answers
40 views

Probability of generating a sequence of numbers between 1 and n

Let's say we want to generate a random number between 1 and n, n-times,then the probability that every integer between 1 and n appears once in the generated sequence of random numbers is $\frac{n!}{n^...
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3answers
78 views

Two aspects of randomness

Consider a random sequence of integers 1, 4, 3, 8, 2, 5, 3, 8 ... The only sufficient condition for the sequence to be random is its unpredictability ie. probability of any number coming next ...
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1answer
841 views

Random numbers correlation

I was doing some random number generator testing when I noticed something about ran1 generator from Numerical Recepies. I generated numbers in range $\left<2.5,7....
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1answer
41 views

Uniform distributed sequences

If $X_n$ uniformly distributed $U[0,X_{n-1}=x_{n-1}]$ for $n>0, X_0=1$. How do we show the sequence $X_n^a$ is uniform.
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2answers
1k views

Calculating first and second moments for random sums?

Assume that $N$ and $X_1, X_2, \ldots $ are all independent and identically distributed over $(0,1)$ with the density function: $f (x) = cx^2 (1 − x)^2$. An integer–valued random variable, $N$ ...
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19 views

Spectrum of sum of (weighted) random matrices

Coming from statistical physics, I am interested in the spectrum of the following sum \begin{equation} \sum_{n=1}^m c_n X_n, \end{equation} where $c_n$ are non-random real numbers and $X_n$ are ...
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1answer
23 views

Complete randomness/disorder and determinism

Wikipedia page about randomness says that "complete disorder" and "true randomness" are impossible according to Ramsey Theory and Cristian S. Calude. I don't understand it. https://...
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1answer
929 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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2answers
50 views

Using Ito calculus to prove that $\int_0^t W_s^2dW_s = \frac{1}{3} W_t^3 - \int_0^t W_s d_s$

I am busy trying to teach myself some stochastic calculus and have come across a statement that I am trying to prove. How can I prove that \begin{align} \int_0^t W_s^2dW_s = \frac{1}{3} W_t^3 - \int_0^...
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1answer
25 views

Series of probabilities

Consider $Y_k=X_1+...+X_k $, where $X_k \in \mathbb{N}_0$ are i.i.d random variables and $E[X_1]<1$ $$\sum_{j=1}^{\infty} P(Y_j=j) \overset{!}{=}1 $$ How can I verify that this equation is true or ...
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1answer
74 views

Benford's law not working?

So I recently came across Benford's law and immediately tried to code it out but the answer I got was rather confusing. I think my code is correct I'm pretty sure it is but the result is just not the ...
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1answer
49 views

Simple Random Walk with equal probability of +1 and -1.

You have 1D random Walk, with +1 of probability 0.5, and -1 of probability 0.5. What is the probability that you will reach +10 but never exceed -5? Attempt: The probability of getting +10 is easy, ...
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34 views

upper bound for expected for special random variable

We throw three times independently perfect dice. For each random realisation of our throwing we have the random vector $$(x_{1}, x_2,x_3)$$ where $x_i\in\{1,2,3,4,5,6\}$ for $i=1,2,3.$ We now consider ...
6
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1answer
133 views

Random Shuffle of Groups

Let's suppose that we have 54 peoples and we arrange them into 9 groups of equal size, so this means that each group will have 6 persons in it. I want to find a procedure, such that the groups are ...
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1answer
51 views

How do we show $P(A) \leq P(A \Delta B) + P(A \cap B) \leq P(A\Delta B) + P(B).$?

In some questions I have been going through here, I came across this inequality several times. $A$ and $B$ are RV. How do we show $$P(A) \leq P(A \Delta B) + P(A \cap B) \leq P(A\Delta B) + P(B)$$.

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