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

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12 views

maximizing over variable with random variable as upper bound

Let X be a random variable. Does the following hold: $$ \max_{g\in(0.2,0.3)}(P(X>g)+P(X\leq g))\leq \max_{g\in (0.2, X)}P(X>g)+\max_{g\in (X,0.3)}P(X\leq g)$$ The separate maximization is clear ...
12
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1answer
163 views

Fitting a parabola to separate two classes of points in the plane

Suppose we have a set of points $(x,y)$ in the plane where each point is either boy or a girl. Does there exists a randomized linear-time algorithm to determine if we can fit a parabola (given by a ...
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2answers
194 views

Monkey typing keys probability formula [closed]

A monkey is sitting at a simplified keyboard that only includes the keys “a”, “b”, “c”, and “d”. The monkey presses the keys at random. How many sequences are there of length eight that use at most ...
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1answer
55 views

What is the formal definition of randomness?

What is the formal definition of mathematical randomness? For example, if one were to prove that a random number generator is random, how would one go about constructing a proof?
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1answer
248 views

When is random selection skewed (untrustworthy)?

Imagine there is a population of 100 people, out of which 3 are to be randomly selected each day for alcohol testing. After a month of such selections (after 20 selections), how many times somebody ...
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0answers
30 views

If you resize a pink noise image, will the pink noise be preserved?

I have a pink noise image of e.g. 500x500 resolution. If I resize the image to a new size with imresize, will the resulting image be pink-noise or will it be something close to it, but not the same? ...
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0answers
28 views

Expectation of product of matrices

Let $H$ be a random matrix, and $x$ be a random vector with $E[xx^{H}]=R$. They are independent. I want to calculate $E[Hxx^{H}H^{H}]$, where $A^{H}$ is a Hermitian of $A$. Does the equation ...
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0answers
14 views

Singular vectors of random Gaussian matrix

Let $A$ be a singular vector matrix of a random Gaussian matrix. The entries of the Gaussian matrix are i.i.d., so the singular vectors are distributed isotropically. Is it possible to get ...
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1answer
18 views

Generating a deterministic random function that takes into account the date as well as a long ID

In SQL Server there's a function RAND() that if you pass a seed to it will return a pseudo random number based on the seed. No matter when you pass that same seed into this function, it will always ...
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0answers
45 views

What does it mean for a sequence of random variables to be increasing?

According to a textbook I'm reading, one statement of the monotone convergence theorem is: If $X_1, X_2, ...$ is a sequence of non-negative random variables increasing to the random variable $X$, ...
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2answers
35 views

What random variable is this?

I have a sequence of reals $S = s_1,s_2,\dots,s_n$ such that $s_i-s_{i-1}$ is a Gaussian distribution. From histogram of sequence $S$ (10000 elements) it appears that it is uniform distribution. Is it ...
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12 views

Prove that the mean of the $u_i$'s generated by a congruential linear generator of complete period is $\frac{1}{2} -\frac{1}{2m}$

Congruential linear generator (pseudo random numbers) I have to prove this proposition but I don't know how. I tried by induction on $m$, the modulus of the generator, but it hadn't got me anywhere: ...
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1answer
52 views

Attempt to solve a matrix (counterbalancing) problem computationally gives “spooky” result: why?

This question is posted on the mathematics section of stackexchange because my uneducated guess is that the answer involves some basic mathematical principles, possibly in the domain of linear ...
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1answer
64 views

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers. [closed]

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers $Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
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1answer
26 views

Random sum of normal distribution with bounds Poisson distributed?

A random variable, $M$, is Poisson distributed with $\lambda=2$. ${X_1,X_2,\dots}$ are independently identically distributed random variables with $\mu=3$ and $\sigma=.2$. Introduce a new random ...
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0answers
18 views

Is there any trick for evaluate this integral?

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
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2answers
39 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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0answers
10 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
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2answers
31 views

What is the probability or percentage or frequency by which hello line will be printed?

I have a below method which is called every one minute from background thread and that background thread keeps running always. ...
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0answers
42 views

Random selection in percentage

I have $3$ lists of things. I want to choose things randomly from this lists. But have percentage limitation on the lists. Like I can select only $60$% things from $1$st list, $30$% from $2$nd list ...
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0answers
16 views

Switching blinding factors securely.

My question is related to information security area and I have asked almost a similar question in: ...
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0answers
55 views

Euclidean norm of two random variables

I have two normally distributed random variables. $X_1$ and $X_2$ with mean $u_1,u_2$ and variance $s_1^2,s_2^2$. They are independent with each other and have interval $(-\infty,\infty)$. Is it ...
2
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1answer
56 views

Measure of card shuffling randomness

I've read online that you need to shuffle a deck of cards at least 7 times (depending on the game being played) for the deck to be 'random enough', i.e. that it is nearly impossible to predict the ...
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0answers
35 views

Is it possible generate all integer number (from 0 to 2147483647 ) randomly without repetition

From a mathematical point of view it's possible do what I wrote in the title? In computer science I could resolve the problem creating an array which contains all value from 0 to 2147483647 and then ...
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1answer
26 views

ideal number gernerator

I was trying to solve a problem on Hackerearth. Here: https://www.hackerearth.com/problem/algorithm/ideal-random-number-generator/ I solved this partially:https://ideone.com/pXkHwQ (passed three ...
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2answers
114 views

Baseball related problem (balls and boxes)

Thanks in advance for any help! So I am trying to figure out if the number of hits an inning of baseball is random, or if hits tend to come in bunches. To do this, I'm just using a fairly small ...
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0answers
27 views

Expected Number of Cycles in Random Permutations with a Random Number of Symbols

It is known that for permutations sampled uniformly from $S_k$ that $\mathbb{E}[C] = H_k = O(\log k)$ (more precisely, $\Theta(\log k)$), where $C$ is the number of cycles in a random permutation. If ...
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1answer
24 views

How would I go about calculating values for distributed percentages?

This might be really simple, but I just can't see it. I'm working on a game that has 8 items each with a (arbitrary) probability (% chance) of being found: ...
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0answers
25 views

Random process theory: probability distribution of height vs summits

Imagine I have a matrix of height values ($z$), e.g. a surface height topography. This surface is a random process: randomly rough isotropic surface with Gaussian distribution. What is the difference ...
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0answers
32 views

Puesudorandom generation

Hi i have created a stream cipher that creates a'random' stream of numbers(1-99) as subkeys for the message.The generation algorithum is as follows:(key mod (iv+counter)) mod 99.The key is the main ...
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2answers
19 views

Probability of Finding Disconnected Patterns in Digit Positions of Random Integer Sequence

Let's say I have a random sequence of 9 digits. x x x x x x x x x What is the probability of finding '9' in the first position and '93' in the fourth and fifth ...
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2answers
49 views

Empirical Process estimation using gaussian density and specific random generator

EDITED: To formulate into math framework: I have a sampling generator producing IID gaussian. To highlight the convergence in the distribution, I calculate the following error. Given a precision ...
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0answers
44 views

The variance of a sum of random vectors

There are $n$ vectors each containing exactly $q$ random variables as elements. Each vector is denoted I$_k$. Each variable within the vector has its own (normal) probability distribution, and the ...
2
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1answer
73 views

Expected number of weighings to get heaviest rock

You are given ten rocks and you would like to find the heaviest one. With each weighing you randomly select two of the rocks and determine which of the rocks is heavier. What is the expected number of ...
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0answers
18 views

Generating random GPS probes

I'd like to generate a certain number of random GPS probes, with the following requirements: The coordinates (latitude and longitude) must be inside a circle with given center and radius. The ...
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0answers
17 views

How to calculate the average of $exp(-t^HA(I+B^{-1}A)^{-1}t)$ w.r.t. A and B.

How to calculate the average of $exp(-t^HA(I+B^{-1}A)^{-1}t)$ w.r.t. A and B, where A and B are Wishart matrices and follow $CWis{h_M}\left( {{\sigma ^2_1}{{\bf{I}}_M},N_1} \right)$ and ...
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2answers
84 views

How to calculate the probability of the next value in a Random Sequence

Assuming that $X$ is the next value in a finite length random sequence, e.g. $$ abbcccaaccbacccabbacababaccccbacabcbacacX $$what would be the $2$-letter most likely to correspond to the value of $X$ ...
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0answers
50 views

Shuffle Deck $7$ times or $8.55$ times?

Persi Diaconis showed in $1992$ that in order to shuffle a deck of $52$ cards, you need at least $7$ riffle shuffles. However, in the paper he published, he showed that we needed ...
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0answers
31 views

Are there any algorithms or formulas to test the randomness of a sequence?

I have a sequence of numbers generated by a programmed computer using pseudorandom functions. Is there a formula which will allow me to see how random this series is? Here is my idea, are there any ...
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2answers
66 views

Pseudorandom Number Generator Using Uniform Random Variable

I am working out of Mathematical Statistics and Data Analysis by John Rice and ran into the following interesting problem I'm having trouble figuring out. Ch 2 (#65) How could random ...
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0answers
26 views

Matching probability of uniform character sequences with known bias

A sequence of characters from a $2^a$ alphabet which are uniformly distributed the expected probability of seeing an exact $n$ character sequence from some starting position (no overlap) is $2^{-na}$. ...
3
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2answers
61 views

How to generate random numbers with Laplace distribution using uniform distribution

I want to prove that transfrom $x=\ln\frac{y_1}{y_2}$ generate random numbers with Laplace distribution where $y_1$ and $y_2$ are unifom random numbers $U \sim (0,1)$. And also transform $x=y_1-y_2$ ...
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1answer
70 views

How can a stopping time be independent of its stochastic process?

I was reading about a special case of Wald's equation, which led me to the following question: If $X_t$ is a sequence of iid RV's, and $\tau$ is a stopping time for this discrete stochastic process, ...
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2answers
84 views

The Random 666 number. [closed]

Does the number 666 more likely to come up more than other numbers like 777? The register, on stock exchange and at the pump? enter image description here enter image description here
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1answer
59 views

How one can be random when choosing numbers in an interval?

My curiosity came from a lottery game. When you are choosing numbers in an interval you tend to choose some numbers that are relevant to you, when the result of the game is shown you think you ...
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0answers
27 views

What probability distribution is being sampled from in this code?

I am reviewing a piece of code where random samples are generated using a function genrand_real3 which gives pseudorandom uniformly distributed floating point ...
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0answers
30 views

How to generate random boundary programmatically?

What should I read to understand how to draw programmatically random oil 'boundary' like on the picture below? Yes, it should go from the top to the bottom and so I don't need these 'long' drops ...
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0answers
38 views

Randomly select vector with bounded norm

I would like to select randomly an $n$-dimensional real vector $\mathbf{x}=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ such that its norm is bounded by a positive real number, say $\sqrt{a}$, $a\in\Bbb{R}_+$. ...
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5 views

Rotation schedule in which members in groups of 7 will work with everyone in the 5 groups across 5 different activities

In our experiential learning center I have 5 groups of 7 people who will move through 5 different activities. I need an assignment method where everyone gets the chance to work with the other 34 ...
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0answers
34 views

Generating random numbers of bell curve distribution

I want to generate random numbers that fit a bell curve distribution. Basicly, I need random numbers from 0 to 1, but I wish to have a high likelihood of it being close to 0.5, but not guaranteed, ...