Questions tagged [random]
Questions relating to (pseudo)randomness, random oracles, and stochastic processes.
1,841
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Brownian motion and logarithm inside averaging
We consider Brownian motion as a process
$$ V(0)=0, \qquad \overline{V(x)}=0, \qquad \overline{(V(x)-V(y))^2}= 2 |x-y|. $$
I do not know how to prove the statement:
$$ \overline{\log\left(\int \mathrm{...
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Why can't the support of an ergodic IRS contain countably many isomorphism classes of subgroups?
An ergodic invariant random subgroup (IRS) ν of a countable group G is called diffuse if ν( { H ∈ SubG | H ∼= K } ) = 0 for every subgroup K ⩽ G.
I'm studying a paper which states "for ergodic ...
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Random Effects vs. Fixed Effects Model? Interpretation of a reduction in coefficient?
Suppose we have panel data and estimate once with the "Random Effects Estimator" and once with the "Fixed Effects Estimator". In the second estimation, the coefficients have been ...
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Linear Congruential Generator has toggling lowest bit?
I am learning about linear congruential generators and read a page at the Lawrence Livermore web site about it. It says the following is a good LCG random number generator:
$$
x[n] = a x[n - 1] + b (...
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Prove or disprove: There is a way to choose independent random chords in a circle so that their intersections are uniformly distributed in the circle.
Prove or disprove:
There is a way to choose independent random chords in a circle so that
their intersection points (given that they exist) are uniformly distributed in the circle.
One common way to ...
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Distribution of distance between eigenvalues for GOE when $n=2$
This question is a possible duplicate of PDF of the Difference of Eigenvalues of a GOE
I was reading Introduction to Random Matrices Theory and Practice by Giacomo Livan, Marcel Novaes, Pierpaolo ...
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Weak convergence of a random stochastic process [closed]
I would like to know if I well understood the notion of "weak convergence of a random stochastic process in $(C([0,\infty[), \left| \right|_{\infty}$.
Is it true that :
Given a sequence of ...
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Powers of Positive Random Variable are monotonic
While reading a paper, I came across the following inequality:
Let $m_1, m_2$ be nonnegative integers, and $X$ a random variable of magnitude at most $1$. Then,
$$E(|X|^{m1})E(|X|^{m2}) \leq E(|X|^{...
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How does the probability density function decay off for a 2D random walk with shrinking step size $f(n) = \frac{1}{n}$
Consider a 2D random walk with the magnitude of the nth step fixed by the function $f(n) = \frac{1}{n}$ and the direction being random.
I know that the root mean square comes out to be,
\begin{...
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Relationship between eigenvalues and positive semi-definite matrices?
I've been trying to write a function (python) to sample covariance matrices. Not sample from them, but to sample the matrices themselves. What I've found is that the positive semi-definite constraint ...
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How do I use math to calculate the percentages of multiple random events happening
I'm creating a mod to a video game that will list the percent chance of an event happening. I know how to get individual percentages, but not a total percentage of the events.
The Problem:
So there is ...
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Adding random noise to an ill-posed problem
Suppose I have found a (finite) solution $x$ to an ill-posed problem, e.g.
\begin{equation}
b = A x
\end{equation}
where $A$ is a $N\times N$ matrix with a large condition number, $b$ is a vector ...
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Show that $\mathbb E[X\mid Z_0,Z_1,\ldots,Z_t]$ is a martingale, where $\mathbb E[X]<\infty$ and $Z_t$ is a martingale.
Let $X$ be a random variable such that $E[|X| < \infty$, and let $\{Z_t: :t = 0,1,\ldots\}$ be a random sequence. We define the random sequence $\{X_t: t = 0,1,\ldots\}$ by $X_t = E[X\mid Z_0, Z_1, ...
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The integral of telegraph process with GWN is a gaussian - having trouble understanding proof
let $N(t)$ be a poissonian counting process with parameter $\lambda$, we'll define $X(t)$ as a telegraph process in the following way: $$X(t) = B \cdot (-1)^{N_t}$$ where B gets values $\{-1,1\}$ with ...
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A shuffling algorithm that limits the number of consecutive repetitions?
This question comes from Stack Overflow. I feel that we need more of a mathematical breakthrough, so I forward the question here.
I also found a similar problem that seems to be a special case of this ...
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Using random hexadecimal characters to generate an even distribution of random numbers within an arbitrary base-10 range
I'm using a random number generator to produce a huge string of random hexadecimal characters which I then cache and pull from to generate base-10 integers within a requested range. The original (...
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Norm inequality for a linear combination of Gaussian vectors
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
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Visual representation of accept/reject sampling
I understand how Accept/Reject sampling works, but sometimes I see graphical representations like this one:
Here we have the sample distribution above the target distribution. Then at a certain x-...
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Small random matrices
Random matrix theory addresses the asymptotic limit, $N \rightarrow \infty$. Are there results for spectral densities of small random matrices, $N \sim O(1)$? I am not interested in the results a-la ...
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Asymptotic integral over rapidly randomly oscillating function
As a part of my bachelor thesis on cosmic structure formation I have been dealing with a sum of many randomly distributed phase factors, so in principle with a pearson random walk. If there are $N$ ...
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164
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Bounding the image size for a function
Consider a bijective function $f$ from $\{0, 1\}^n$ to $\{0, 1\}^n$.
Now, sample a random $k$ to $1$ function $g$ (that is, from the set of all possible $k$ to $1$ functions from $\{0, 1\}^n$ to $\{0, ...
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Percentage of non picked balls with replacement
Experimenting with numpy and generating random arrays of integers (discrete uniform distribution) I noticed something but can't explain how it would be calculated.
I generate a random sample of ...
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What is the chance of getting same decimals of percentage 4 times in a row with RNG?
I was rolling some random percentage with RNG between 0 and 1; if you multiply the result to 100, you're getting the percentage between 0 and 100, decimal amount was 4. And then I got 0.7171% the ...
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A question in operator norm of random matrix by epsilon net in Tao's RMT notes
I am reading Tao's Random matrix notes, specifically Proof of Corollary 2.3.5.
Corollary 2.3.5. (Upper tail estimate for iid ensembles). Suppose that the coefficients $\sigma_{ij}$ of $M$ are ...
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Rolling a die, then rolling another die for unique results
I will roll a fair 100-sided die. If I get a 1 or 2, I will roll a 40-sided die.
How many times will I have to repeat this experiment until each roll on the 40-sided die appears at least once, on ...
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1
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56
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Probability prediction verification
I recently came up with an idea of this problem, spent some time trying to solve it, and I'll appreciate your help finding a solution :)
A scientist conducts an experiment in quantum physics. The ...
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Exchangeability of sequence of RVs
I have the following question.
If I have the following setup: $X_i=F(\xi_i)$ for $i\leq N$, i.i.d. RVs $(\xi_i)_{i\leq N}$ and some measurable $F:\mathbb{R}\to \mathbb{R}$, and $(Y_1,...,Y_N)=G^N(\...
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Expected value of normalized random variable [closed]
I have $N$ random variables $X_1, \dots, X_N$ that are independent and identically distributed. Define the quantity $y_i = \mathbb{E}\left[ \frac{X_i}{\sum_{j=1}^N \alpha_j X_j}\right]$, where $\...
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How to prove the tail bound of summation of non-symmetric Bernoulli random variables?
Let $a_{ij}$ as the elements in adjacency matrix of Erdos-Renyi graphs. Thus $a_{ij}$ is a Bernoulli random variable:
$a_{ij}=1\text{with probability} p$, or $0\text{with probability} 1-p$
Now I would ...
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What is the operator norm of a function of n multiply a gaussian matrix?
Let $W$ be Gaussian Wigner matrix, and it is well-known that its operator norm satisfies
$$P(\|W\|\leq 2\sqrt{n}+t\geq 1-2\exp(-ct^2)).$$
My question is too basic but I cannot figure it out:
What is ...
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Product of random variables is uniform
Require a random variable $X$, $0 < X < 1$ with a distribution such that the product of two independent samples from the distribution is uniformly distributed on the interval (0,1).
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Probability mass function for picking objects
I have $N$ objects, each having a performance function that depends on $|X_i|$, where $X_i$ is a "complex" random variable and $|\cdot|$ denotes the module of a complex number. The ...
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Autocorrelation and Power Spectral Density for Wide-Sense Cyclostationary Processes
I am reading in a book called "Understanding Jitter and Phase Noise" and came across the following equation and need a little help to understand his justification for a certain step in the ...
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1
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How long would it take to land on $\frac{1}{10^8}$ if a random number was selected every second?
I am making a program to basically pick a number from $1-100$ million per second I would like to know how long would it take to land on $1$. I could just run this program to see but I predicted that ...
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Randomized algorithm to solve “needle in a haystack” problem?
In the book "Introduction to quantum algorithms via linear algebra" they said that:
"The problem that Grover’s algorithm solves is finding a “needle in a
haystack.” Suppose that we ...
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Deranged generous gift giving in the limit
I was writing a routine for a game and stumbled upon some algorithm issues and a theoretical question about it. Here I'm mostly concerned with the latter, which may be phrased as a sort of graph ...
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$\mathrm{E}\left(\int_0^ts\mathrm{d}(B_s^2)\right)^2$ [closed]
Here is a practice problem I encountered with Stochastic integral expectation. How to use Ito's formula to calculate the expectation of the following stochastic integral?
$$
\mathrm{E}\left(\int_0^ts\...
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Covariance of functions of two random variables
I am trying to show the following: Suppose that x and y follow identical and independent distribution. F(x, y) is an increasing function of x and y. Is there any way to show that the covariance ...
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Probability of a pair in "strip" card shuffle. [duplicate]
I was playing around with a deck of cards and I wanted to try out what the odds for a identic pair of cards during a strip-like shuffle were. What I mean by strip like shuffle is the shuffle of the ...
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Definition of the convergence in probability of a random measure
I'm kind of confused about the definition of convergence in probability of a random measure $\mu_{n}$ to a deterministic limiting measure $\mu$, which are probability measures on $\mathbb R$.
Question:...
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Randomize/shuffle sound tracks to play each one for a preset portion
Suppose you have $n$ sound tracks, $n_1, n_2, \ldots, n_n$. Each has a duration of $x_i$ minutes, so track $n_1$ is $x_1$ minutes, $n_2$ is $x_2$ minutes,... etc. You pick one track by a randomizing ...
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Expected number of attempts to generate random binary sequence of length N containing at least x 1's
I'm generating random binary sequences of fixed length N
Example with N=10: 0101011100, this one has x=5 1's
How often do I have to generate such a sequence until I get one with at least x 1's on ...
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Estimate the average difference from the expected amount of bit-runs in true random data?
I'm creating blocks of totally random bits using a TRNG and per X amount of bits I count the number of times "bit-runs" of different lengths occurred.
With bit-run I mean sequences of ...
2
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Why does the distribution of entropy in a deck of shuffled cards not follow a normal distribution?
Around a year ago I did a study on shuffling cards and came across an unexpected result. I was investigating methods for quantifying how "shuffled" a deck of cards was, such as measuring the ...
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Algorithm implementing multivariate normal distribution from Sobol QRNG
My goal is to implement a mathematically correct multivariate normal distribution using Sobol QRNG sequence as a source of randomness. The implementation should NOT produce the whole set of a given ...
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37
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Probability of Random Integers in a series of ascending/descending Runs
Suppose you have a set of $N$ random integers. Each integer is limited to the range from $0$ to $99$.
You then consider each subsequent neighbor as "ascending" or "descending", ...
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Threshold for the "number of UUIDs generated per millisecond" at which the collision probability of UUID v4 and UUID v7 is equal
I post this question here instead of StackOverflow because the mathematical element is stronger than engineering one. First, let me clarify the definition of terms.
UUID v4: Random value of $122$ ...
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Randomness of rightmost bits from LCGs
Task: Show that the sequence of integers made up of the $k$ rightmost bits generated by an LCG with $m = 2^n$ has a period of at most $2^k$.
I actually get a hint to define the output as the following:...
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Is sampling (x,y) from a bivariate gaussian distribution the same as sampling x and y each from their own univariate gaussian?
Suppose I want to randomly sample (x,y) from a bivariate gaussian/normal distribution:
$$f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y} \exp\left(-\frac12\left(\frac{x^2}{\sigma_x^2} + \frac{y^2}{\sigma_y^2}\...
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If we have a coin toss, would this proof be considered as an Almost sure convergence proof?
If we have $n$ coin tosses and we want to prove that the coin will eventually land heads, will this proof be considered a proof of a.s convergence?
Let's consider a contracting sequence of events $A_1,...
