# Questions tagged [random]

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

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### How to quickly determine if A - (A ∩ (ΣBi)) set exists and pick a random value from it?

I am trying to make my python code randomly and quickly find a point (x,y) in the A disc and that is not part of any other present discs, if that point does exist. However, the only method I have ...
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### Sum over random index set

I'm having a hard time finding literature concerning the following problem: Let $(\Omega, F, \mathbb{P})$ a probability space, $n\in \mathbb{N}^{\star}$, $(x_1,...,x_{n})\in \mathbb{R}^{n}$. Define ...
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### Number of Inversions in Inverse Transform Sampling from a Distribution

I understand the process of inverse transform sampling, but one component doesn't make sense to me: The wikipedia article states: In order to obtain a large number of samples, one needs to perform ...
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### How can we determine if a random variable $\limsup _{n\rightarrow \infty } \left( \prod ^{n}_{k=1}X_{k}\right) ^{1/n}$ is measurable?

I would like to prove that $\limsup_{n\rightarrow \infty }Y_n$ is measurable when $Y_n$ is defined as $$Y_n = \left( \prod ^{n}_{k=1}X_{k}\right) ^{1/n}$$ for a sequence of positive random ...
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### Random generation of Hermitian matrices

I have developed an algorithm to generate normalized Hermitian matrices, given the dimension N, the maximum number of non-zero elements in each column d, and the condition number k, that is, the 2-...
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### Lower bound on the singular values of a matrix with uniformly distributed random numbers

Given a matrix $A\in\mathbb{R}^{m\times n}$ whose entries are randomly generated and uniformly distributed in $[0,1]$, is there a formula for a lower bound on the singular values of $A$? (Of course ...
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### What is the probability that 4 random bytes all increase in value?

From a randomness test @ https://www.vmpcfunction.com/c6.htm (2/3rds down the page):- Four-Byte-Run-Up Test The sequence is divided into four-byte segments. If the first byte of the segment is less ...
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1 vote
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### Probability distribution of exponential chickens

This problem originates from minecraft chickens. Let's say I start with one ( fully grown ) chicken. This chicken lays one egg every $\left[5, 10\right]$ minutes, ...
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### Order given by two probability measures

Let $X$ be a r.v. and two measures $\mathbb{P}$ and \mathbb{Q} such that $$\mathbb{E}^\mathbb{P}[X]\leq \mathbb{E}^\mathbb{Q}[X]\quad (1)$$ It implies \...
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### The probability of a lottery eventually wins

Suppose there is a gamble, and there are two lotteries. If in each round, only one of these three cases can happen: (1) lottery A gets one score, with the probability of $1/3$; (2) lottery B gets one ...
1 vote
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### Exercise 3.3.3 (c) of High dimensional probability by Roman Vershynin

I have question Exercise 3.3.3 (c) from Chapter 3. Exercise 3.3.3 (Rotation invariance). Suppose $G$ is an $m \times n$ Gaussian random matrix, i.e., the entries of $G$ are independent $N(0,1)$...
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### Understanding the derivation of the expression

in my research work related to wireless communication, I have the following expression: $$\tag{1} r_1 = \sqrt{P}H_1H_2(\textbf{h}_2^T\Phi \textbf{h}_1)s+\sqrt{P}H_0h_0s+w$$ where, $r_1$ is received ...
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### Selecting a random set of 10 numbers between 1 and 100, what are the odds that some subset add to 100?

Selecting 10 numbers between 1 and 100 without replacement (i.e., such that the selected numbers are distinct from each other), what are the odds that some subset of those numbers add up to precisely ...
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### Linear combination of vectors with Gaussian coefficients

This is a cross-posted question of here. Since I have trouble of formulating the problem formally, I wish to receive some feedback/help from Stackexchange community. My question Motivated by a simple ...
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### Conditional probability for 3 random variables failing

I'm sure I'm doing a wrong assumption here but I can't see it and was hoping someone would point it out. Suppose three continuous random variables $x_t, x_{t-1}$ and $x_0$. We can write the ...
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### What is the probability of $\alpha$- quantile

I didn't understand why the $P(X \leq q_{\alpha}^- (X))=\alpha$ for $0<\alpha<1$. Please help me with this. The setup for it: X is a real random variable and the lower $\alpha$-quantile defined ...
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### what is probability density and distribution of X?

We have a relation, $0 \leq X \leq Y$. where X and Y are random variables, and we know the $$Y \sim Exp( \eta )$$ I need to find the probability distribution and density of X. What I tried below, ...
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### a basic question on random process

I am currently studying the infimum of a random process, however, it seems we can study it according to its supremum, but I am not sure my naive thought was correct or not. Speficially, consider ...
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### Lumbroso "dice roller", correctness of the algorithm for 32 bit integers [closed]

In this paper Optimal Discrete Uniform Generation from Coin Flips, and Applications Lumbroso develop a nice algorithm: The FAST DICE ROLLER algorithm; it returns an integer which is uniformly drawn ...
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### Aggregates and Psuedorandomness

Psuedorandomness in compsci usually takes the form of Math.random — in binary it is theoretically possible to get a random-looking number from the method: Math.random(MaxInteger) is expected to return ...
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### Picking random natural number

It is not possible to pick a random natural number (out of all natural numbers), such that each number has the same probability. But is it possible to define a probability distribution over the ...
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### Rank of binary matrices: {0, 1} vs {-1, 1}

Let's say you have two matrices, $M_{\{0, 1\}}$, a binary matrix with entries in $\{0, 1\}$, and $M_{\{-1, 1\}}$, a binary matrix with entries in $\{-1, 1\}$. Both matrices are identical, except where ...
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### Define this step function over the rational numbers

In desmos I plotted a step function (I only plotted 30% of it). Here is my graph: This function is a function from $\Bbb Q\cap (0,1) \to \Bbb Q\cap(0,1).$ The step function is generated by counting ...
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### Derivative of Multivariable Function of Random Variables (as functions of time) with respect to Time

Consider the following function: $$F(t, X, Y)=V(t)\cdot W(X, Y)^{\alpha}$$ Where $X$ and $Y$ are random variables that change continuously over time $t$. Function $W(X,Y)$ is function of those two ...
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### How to "predict" LCG-based pseudo random numbers?

Note: By entering the caption math.SE listed a bunch of posts but none answerd my question. Also Wikipedia was no help for my case, so I dare to ask here. The LCG $s_{k+1} = a\cdot s_k \pmod m$ in ...
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### Miller-Rabin primality test and random

Miller-Rabin primality test need random choosing number in each round. Time of random() is negligible compared to time of round ? Problem is, how to generate big random numbers? Maybe best would be ...
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### Probability that a uniform random variable + a constant be greater than another uniform random variable

I am trying to understand a model of probabilistic voting from the following paper https://ideas.repec.org/a/eee/pubeco/v96y2012i1p10-19.html and the one component I struggle with goes as follows: ΔU ...
1 vote
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### Bounding absolute value of centered row sum of adjacency of Erdos-Renyi graph

I am trying to understand the formula (5.14) in this paper, explicitly, the goal is to upper bound $$\sum_{i\neq j}|a_{ij}-p|$$ where a_{ij}=\begin{cases} 1,\text{ with ...
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### How to bound a sum of absolute value of centered Randemacher variables

Consider $$A_{i}=\begin{cases} 1,\text{ with probability } 1-p\\ -1,\text{ with probability } p\\ \end{cases}\,.$$ and $p<1/2$. What would be the ...
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### Period Length Of A Multiplicative Congruential Generator (Carmichael Function) [duplicate]

First: By entering the caption math.SE listed a bunch of posts but none answerd my question. So I dare to ask: Taking a closer look to the ancient HP-21S pocket calculator reveals the nature of its ...
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### Transforming numbers into fake randoms that are actually deterministic

How can I convert a number into a seemingly random number in a process that is actually deterministic? For example I'd like to transform: 123456 into ...
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### The Facebook Birthday Problem（Birthday Problem Variation） [closed]

The Facebook Birthday Problem: This problem stems from the classic Birthday Paradox. It says: How many friends do you need for the probability of having at least one friend with a birthday each day ...
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### Comparing One-Step vs. Two-Step Random Selection Methods in a Code Golf Challenge.

One-Step I'm reviewing answers to this code golf challenge. Here's a target cell marked by X that was chosen using rand()*512 (...
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### Lotto Scam or Misunderstanding

a state lottery offers a came with the following setup Choose five (5) numbers from (0 through 9) or use the Quik Pik option and let the computer select for you. You win by matching your numbers to ...
1 vote
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### How to construct a maximum-information embedding of sampled objects using a binary function?

This feels like a very specific problem, but I hope there already is a method to achieve what I want. There is a random process from which I can draw samples of non-numerical, variable sized objects (...
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### Average number of random integers needed to satisfy any modular subset sum?

A multiset $S$ of numbers is considered satisfactory if $\left|\left\{\sum U \bmod n \mid U\subseteq S\right\}\right| = n$, that is, all remainders mod $n$ can be constructed by summing subsets of $S$....
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### How to perform a random split on a value

I have 30 dollars, randomly split it into 3 parts, and send it to persons A, B, and C. After playing an infinite number of games, the expected payoff of each person can be: Case A ...
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### Analytical solution to marble toss experiment

I am interested in the following problem: Assume we have an infinitely large, homogeneous grid, where each grid point is connected to 8 neighboring grid points (top/bottom, left/right, diagonal). Now ...
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### Can you seed a seekable prng in order to force it to generate a given subsequence of values?

Melissa O'Neill, (in her PCG prng papers), makes reference to prng "party tricks" where she says that you can seed a prng appropriately in order to force it to generate a sequence that ...
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### Does $E(X_n)=\infty$ imply that $X_n/n \rightarrow 0$ in probability?
I am struggling with an exercise… the first part of the exercise was ok, but the second part is a lot less trivial. $X_n$ are non negative iid random variables. First part: check that if $E(X_i)$ ...
### Sequence of independent $X_n$ with $E(X_n)=0$ but sample mean diverging to minus infinity [closed]
I am asked to construct a sequence of independent random variables $X_n$ with $E(X_n)=0$ for any $n$, but the sample mean of the $X_n$’s diverges almost surely to minus infinity. I understand that the ...