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

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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
50 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 ...
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1answer
75 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
19 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
18 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 $CWis{h_M}\...
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2answers
97 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
52 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 $\frac{3}{2}\log_2n$...
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0answers
32 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 ...
3
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2answers
71 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 $2^{-na}$. ...
4
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2answers
77 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 $x=y_1-y_2$ ...
2
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1answer
79 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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1answer
92 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
2
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1answer
67 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 wouldn'...
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0answers
28 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
31 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
39 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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0answers
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
54 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, ...
3
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2answers
36 views

Are there different degrees of “random” or is it exclusively a binary result?

I'd like to understand if the randomness of a sequence is boolean or if there are different degrees of randomness. Perhaps the randomness (confidence) test of a series can be based upon the ...
4
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0answers
63 views

What is the difference between random variable and uncertainty?

Please help me to find the difference between random variable and uncertainty? Can use the formula for the area of random variables (expectation, derivation,...)in the field of nonrandom variables (...
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0answers
26 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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0answers
14 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
74 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
19 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 $i\in[1,M-W+...
3
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1answer
85 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 solved,...
2
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2answers
63 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: $$\Pr(X=a)=\...
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1answer
23 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 we'...
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1answer
32 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
26 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 $p_1^{(i)},p_2^{(i)},...
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2answers
31 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
17 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
755 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
35 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
36 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
30 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
38 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
29 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 $c_1,...
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0answers
37 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 ...
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1answer
30 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 ...
1
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1answer
21 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
38 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 N(\...
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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 ...