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

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25 views

Stochastic process $X(t)=W_tX(t-1)$ with $\left\{W_t\right\}_{t=1}^n$ iid row stochastic matrices

I have been struggling for a while with the following problem. Consider a sequence of iid row stochastic matrices $\left\{W_t\right\}_{t=1}^n$ and the linear dynamical system $X(t)=W_tX(t-1)$ with ...
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0answers
33 views

Can someone give examples of where deviations from pure randomness can be handled by number theory?

In his excellent book, How Not to Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg makes the following statement on page 142: There are some deviations from pure randomness whose ...
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0answers
42 views

What's the name of this extremely common but extremely pathological continuous function?

Okay, so let's define a random function $F$, such that the value of $F(x)$ is uniformly distributed on $[-1,1]$, and such that for any $x$ and $y$ with $x \ne y$, $F(x)$ and $F(y)$ are independent. ...
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0answers
16 views

Generate a random variable by composition method?

I want to generate a random variate by the density function $f(x) = 1.5 \cdot x^2$, $-1 \leq x \leq1$. If i use $x=0$ to divide the pdf into two parts, the process is as follows. But $f_2(x)$ can be ...
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12 views

Use composition algorithm to generate random variates?

How to use composition algorithm to generate random variates whose cdf and pdf is as follows. CDF and PDF
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2answers
73 views

Independent random variables: Sum

Suppose that for $n\in\mathbb{N}$, $(Y_1,\ldots ,Y_{n+1})$ is a finite collection of independent random variables, does that imply that $Y_{n+1}$ is independent of $Y_{1} +\cdots+Y_{n}$? And if so, ...
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0answers
18 views

Expected number of row operations of progressively populating a band matrix

Consider populate an $M\times M$ matrix $\mathbf{A}$ with randomly incoming length-$M$ row vectors: each row vector may have non-zero elements only at indices $i,i+1,\ldots, i+W-1$, where ...
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1answer
79 views

lim sup and lim infs of Brownian Motion: $B_t/\sqrt{t}$ as $t \to \infty$ or as $t \to 0$.

Below is my question. Q7.9 is what I'm stuck on. I've done Q7.8; I included it in the picture because I'll use it in Q7.9, and it gives a definition that I'll use. Update: This question is now ...
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2answers
52 views

Non-integral-over-a-point proof that the probability of any point in a continuous distribution is zero

My Question For continuous random variables / continuous distributions, it is defined that the probability of any point has probability $0$. The most common proof for this is as follows: ...
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1answer
22 views

Probability of an aleatory experiment conducted $n$ times

What is the probability of an aleatory experiment conducted $n$ times? For example, say we choose randomly a number $x$ from a known interval, which happens to contain some certain kind of numbers ...
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1answer
29 views

Extinction time of contact process on finite lattice

Suppose we have a contact process on a finite lattice. I'm asked to give a heuristic argument for the fact that the extinction time for the contact process is exponential in the size of the lattice ...
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1answer
24 views

Mean and variance of a Lagged Fibonacci Generator [closed]

I have stumbled upon this very interesting question. Given an additive LFG: LFG(n) = LFG(n-k) + LFG(n-j) mod m, with LFG(0) ... LFG(k) given What could be the ...
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0answers
16 views

Sampling sequences of random vaiables without replacement in a weighted fashion

Let's we say have $n$ random variables $x_1,x_2,\dots,x_n$. Assume for any random variable $x_i$ the set of all possible outcomes is $\left \{a,b,c,d\right\} $with probabilities ...
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2answers
29 views

Odds of first draw vs last draw

I have debated with myself for a while as to whether, when doing a random draw for multiple prizes, should the 'major' prize be drawn first or last? I would (as a consumer) expect that the odds or ...
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0answers
16 views

Random selection balls into two groups and show they did not change in two trials

I have $10$ balls which are numbered from $1$ to $10$. Randomly select (uniform) these balls into two groups (number of balls in two groups is different). Then compute the mean value of numbers on ...
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3answers
747 views

Is it necessary that if a limit exists at a point it should be also defined at that point?

Say there exists a limit $\lim_{x \to x_0}f(x) = L$. Is it necessary that $f$ be defined at the point $x_0$ itself? Well, what I think of it is that it's OK to be undefined at that point because I ...
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1answer
28 views

Mistake in generating random numbers - no irrational ones

Hi I just wondered if the probability densities have to be corrected when using them on a PC since the number representation is not at all continuous. So we cant simulate any irrational numbers and ...
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0answers
7 views

unltra-simplistic linear congruential random number generators

The real linear congruential number generators are more complicated but it is possible to generate a sequence that hits every number in 32 bits using only: ...
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1answer
35 views

Total possible Combinations

I have a quick question. I'm terrible at math so when I read the other posts on this it made no sense to me. I'm creating a trading card game, and I need to know how many possible UNIQUE cards my card ...
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0answers
28 views

Mean absolute distance for a symmetric random walk

I have found that the mean absolute distance for a symmetric random walk after n steps can be computed using this product: What can be deduced from this? For example how can variance be computed ...
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3answers
42 views

Probability of card color

Task: In box are three cards. First card is red on both sides. Second card is black on both sides. Third card is red on one side and black on other side. I random choose one card and I see that ...
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1answer
37 views

Generating random groups satisfying certain conditions

I am trying to find a counterexample to a claim about groups. I need a random generator (a program in any programming language, or an algorithm) which will generate a random group $G$ of any order ...
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0answers
27 views

Prove the bounds on expected value of minimum distance

Let $X_1,X_2,...,X_n$ be $n$ i.i.d. samples from the uniform distribution on the unit square $[0,1]^2$, and let $x \in [0,1]^2$ be an arbitrary fixed point. Prove that there exists some constants ...
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0answers
34 views

Does Chaitin's constant mean true randomness exists?

I've been reading about Chatin's Constant, and some of the information there seems to contradict what I've heard before. It says that the digits of Chatin's Constant can not be computed. This means ...
1
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1answer
24 views

Is an event with a fixed number of outcomes considered random if one outcome is more likely than another?

My friend says that even though an event has (for example) three outcomes with outcome A at 10 percent, outcome B at 10 percent, and outcome C at 80 percent that the outcome of the event is still ...
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1answer
20 views

The rand function/algorithm - when does it begin to develop a pattern?

this question is rather general but I am sure a specific answerer or at-least a theoretical answerer can be provided on it. The rand function is a random number generator that runs on a seemingly ...
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0answers
37 views

Joint probability density function of two dependent gaussian variables

I have faced the following dilemma while doing some research, so your kind support will be highly appreciated. I have the following two random variables: $$X_1 \sim N(\mu_1,\sigma^2_1)$$ $$X_2\sim ...
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0answers
27 views

What does it mean that a Monte Carlo (MC) algorithm is correct with probability at least $p$?

I was trying to understand the definition of what it meant to say that a Monte Carlo (MC) algorithm succeeds with probability at least $p$. Usually in probability I would have concluded that the ...
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0answers
26 views

What class of probability distributions do probabilistic turing machines induce?

What class of probability distributions is induced by the class of probabilistic turing machines? Is there a precise characterization?
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0answers
12 views

How to generate X random winners from a lottery for a given period of time?

I need to determine lottery winners in the following game-type setting: You can register in the lottery with your unique id (let's say your email address) which means that you can register only once ...
0
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1answer
56 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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1answer
57 views

Random walk - Markov chain

I have a problem.If we start at place $0$ and the probability to go right is $p$ and the probability to go left $q$. I need to calculate the probability after 100 steps that the maximum place when we ...
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0answers
24 views

rms width calculation

I have a Equation with following form: $$v(t)=e^{-( t/T_0 )^2}\sum_{n=-N}^{n=N}C_nCos(n\Omega t)$$ where $C_n $ is uniform random variable that take two integer 1 or -1 ,and $\Omega ,T_0, N $ are ...
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0answers
20 views

Generating a sequence of random numbers as a random seed for gambling

We have a simple game where a player wins if a random number is greater than some threshold (say 0.5 on 0-1 scale). The player commits to a game first, and then we generate proceed to generate the ...
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0answers
15 views

Working out the average number of selections required to select each item of a list at least once when using random selection with replacement?

So, most combinatorics I've looked at while researching this consist of limiting the number of selections made from a set X and working out the probability of a particular combination of results. In ...
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0answers
98 views

product of two chi-squared distributed variables

What is the expected value of the product of two chi-squared distributed variables (E[XY] where X and Y are chi-squared distributed random variables)?
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1answer
14 views

Probability of choosing $x$ in a sample of $m$ choosen from the total of $n$

Sorry for being ambiguous but the question is that a class contains $25$ boys and $15$ girls.   What is the probability of choosing $5$ students at random and choosing $3$ boys? So the $5$ ...
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2answers
171 views

Statistical probabilities to win the lottery

I'm trying to figure out what is the statistical probabilities to win the lottery. Let's assume that we pick 6 numbers out of 40 so : 1/40 × 1/39 × 1/38 × 1/37 × 1/36 × 1/35 = 1/2763633600 Up until ...
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1answer
60 views

Randomic distribution

Suppose I have X elements [A,B,C,D,...] and Y subsets of X of max length 4. Suppose the subsets are build up randomly, is there a way to compute the average number of subsets an item of X is an ...
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1answer
27 views

Need way to statistic puzzle

Hello guys i need help with statistic problem. The problem is : In room have 3 doors, one door is exit from room (o minutes) , second is return us to the room after 3 minutes , and last door is ...
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1answer
28 views

Using dieharder to certify randomness of a random sequence

I would like to prove (at a certain level) the randomness of a random sequence of N bits. I know that the dieharder battery of tests is supposed to be used to test the generator, not a particular ...
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0answers
17 views

Distribution using the Poisson Point Process model

I need to use the Poisson Point Process (PPP) model to randomly distribute a set of 'objects'; over a given area: Let's say that we have N objects to distribute over an area that has been split ...
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0answers
51 views

Random Numbers and probability

How long does a sequence of random decimal digits (0, 1, 2, ..., 9) need to be before you can "reasonably" expect the sequence to contain all numbers from 0 through 999 (inclusive). -- It's up to ...
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0answers
22 views

bounding absolute fluctuations of random variable distribution

I have a discrete random variable with values $\underline{y} = (y_1, y_2, \dots , y_d)$ with corresponding probabilities $\underline{x} = (x_1, x_2, \dots , x_d)$ and average $Y = \sum_i x_i y_i$. ...
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0answers
44 views

Logic behind Metropolis algorithm

I am using Metropolis algorithm to make a program for Ising model in Statistical Physics. In Ising model, we take a collection of spins with initial energy, say $E_i$, then we randomly flip one of the ...
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1answer
29 views

Is there a way to generate this class of bivariate distributions from two uniform random variables?

I need to generate a pair of random variables $U,V$ with distribution $$ f_{U,V}(u,v)\propto(u+v)^{-4} $$ in $u,v\in[1,L]$ with $L>1$. I managed to invert the CDF of the marginal distribution for ...
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1answer
59 views

What is the difference between Discrete and Continuous Random Variables?

What is the difference between Discrete and Continuous Random Variables? Discrete Random Variable = takes countable or discretely determinable values. So, it is always listable. For example, digital ...
2
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2answers
38 views

Why a shuffling algorithm is not uniformly random between [1, N)? [closed]

I've been taught that simple shuffle algorithm will not be uniformly random if the interval for the random numbers is not [1, N), i.e. it can't be from 0 to N, it must be from 1 to N-1. I'm trying ...
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2answers
90 views

Continuous Random Variable question, Probability and Statistics

a little help please A couple decide they really want a daughter. So, they decide to start having children and continue until they have their first daughter. Assuming having either a boy or girl is ...
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1answer
48 views

Convergence of sequence of random variables to infinity

Suppose $X=X(\alpha)$ is a sequence of non-negative continuous random variables indexed by a continuous parameter $\alpha$ such that $\alpha$ is in $[0,\alpha_0)$ for some finite positive ...