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

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13
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4answers
5k views

Is there a “most random” state in Rubik's cube?

Is there a state in Rubik's cube which can be considered to have the highest degree of randomness (maximum entropy?) asssuming that the solved Rubik's cube has the lowest?
2
votes
1answer
414 views

Compressing random numbers

I've been thinking about ways to compress the output from a (supposedly) random number generator. Let's assume for a moment that my computer can produce high-quality random numbers. I'm certainly not ...
2
votes
0answers
19 views

Why could we observe trends in (pseudo-)random graphs?

In finance, markets are evolving from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data, so they are stochastic by nature. ...
5
votes
1answer
968 views

Mapping random points on a sphere onto a uniform grid

Say I had an arbitrary sphere that is covered in a uniform triangle mesh of N elements with each element having a unique sequential index. If given the coordinates of a random point on the surface of ...
0
votes
0answers
7 views

Why %RSD of execution times, while sorting hundreds of arrays, is lower for larger arrays of random integers?

I am experimenting with the sorting of arrays and their execution times. While using bubblesort, insertsort and ...
0
votes
0answers
21 views

random binary sequence [duplicate]

Suppose we generate a sequence containing only 0's and 1's. We generate this binary string by randomly adding a 0 or 1 one at the time. We stop when the sequence contains k more 1's then 0's or k more ...
0
votes
0answers
48 views

Random walk visiting $k$ distinct points

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...
3
votes
1answer
524 views

Fast generation of Pareto-distributed randoms.

I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
7
votes
0answers
151 views

How to attack universal hash function based on finite-field arithmetic?

As per the Recursive n-gram hashing is pairwise independent, at best paper, I want to use the algorithm described in chapter 6 and 7 (page 7 - 10). The hash works as follows: Define a random ...
1
vote
0answers
35 views

discrete random variable with uniformely distributed random variable

I hope you can help me because I have no clue where to start: Let X be a discrete random variable with $ p_k=P_X[X=x_k]=p(x_k) $for all $1\le k\le N$ for $N\in \Bbb N$ and distribuition function: ...
0
votes
1answer
30 views

Does this set of infinite binary sequences have positive probability?

The AMM article "What is a random sequence?" argues (at the end of Sec. 2) that if, from the set of all binary sequences, we remove those (countably many) that have "computable regularities", then the ...
2
votes
2answers
60 views

Random and non-computable numbers

Let $\alpha \in (0,1): \quad \alpha=0.a_1a_2\cdots a_n \cdots \quad$ where the $a_n$ are numbers generated by a physical generator of genuinely random numbers (if it exists). Than it seems that ...
0
votes
1answer
18 views

Probability of maximum of a random walk?

Let us consider a random walk denoted by Sn and let Mn be the maximums of the random walk. Now let us consider that this random walk will end at some point k. SO I am stuck how to prove this equality: ...
3
votes
0answers
44 views

How to punish a riffle shuffle?

It is common knowledge that for a deck to be considered randomly sorted, at least seven riffle shuffles should be used. However, in my experience, very few people will take the time to complete seven ...
-4
votes
0answers
40 views

Simulating random die toss [closed]

Starting with realizations of $Y$, how would you form $T$, which is the RV that simulates the result of a fair die toss that equals the number of dots showing on the top face? Sketch the PMF of $T$.
2
votes
1answer
243 views

What is a thorough method to manually generate a random number between $1$ and $100$?

The other day, I got an idea. I would like to generate a random number between $1$ and $100$ , however by hand. And only using simple tools like a desk clock and pen and paper. What might be a way ...
0
votes
0answers
20 views

Probability of the n-th dice matching in 2 groups of sorted random numbers

Say we have 2 groups of 6 6-sided dice. Each group of dice is rolled and then sorted so we have 2 groups of sorted numbers. What is the probability of each die in the group matching the corresponding ...
1
vote
1answer
397 views

Measuring degrees of randomness

Imagine, for simplicity's sake, that we have a set of numbers, each equal to either 0 or 1. Let's call each a bit. Rationally, if the set is completely random, and reasonably large, the probability ...
0
votes
1answer
32 views

Forced probability using a random number generator, mathematically correct or not?

What might happen if when using a random number generator, if a particular run is not so random, you "help it out" a little bit? For example, if we use a random number generator to simulate $10,000$ ...
1
vote
0answers
32 views

Is a “deterministic” subset of a random subset random?

Let $S$ be some set and consider $X \subseteq S$ of size $|X|=x$ u.a.r. (among all the subsets having this size). Now, use some properties of this set $X$ to find some subset $Y\subseteq X$ of some ...
0
votes
0answers
19 views

Arbitrary vs. random subsets: computing probabilities

Let $G=([n],E)$ be a graph having minimum degree $\delta(G) \geq (1-\delta) n$. For some $q=q(n)$, let $G_q=([n], E_q)$ be the random subgraph of $G$ obtained by deleting each edge independently with ...
2
votes
1answer
96 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 ...
3
votes
1answer
472 views

Probability distribution of the product of two independent complex gaussian random variables

I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that ...
1
vote
3answers
88 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
3
votes
1answer
26 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
1
vote
1answer
48 views

Is there a statistical measure of bitwise entropy?

(Somewhat inspired by this website, particularly Section III. Also, I might be using a different definition of entropy than usual; what I am using is closest to the physics definition (the one I ...
0
votes
1answer
10 views

Extending Random Number Ranges

I am provided with a random number $\xi \in [0,1]$. I check if a particular $\xi_i \lt x$ is true and if so, I need to convert those random numbers within the range $[0,x)$ into uniform range in ...
0
votes
0answers
53 views

Particle in a box

Say I have a point particle located at the center of a box and imagine that I give it a velocity v in some direction. It will bounce back and forth in different directions maintaining the same speed ...
0
votes
0answers
47 views

Sequence of non-independent coin tosses

Suppose that a sequence of coin tosses is due to be performed. Let $p_i$ denote the probability that the $i$th coin toss lands on Heads and let $X_i$ denote the corresponding indicator random variable ...
3
votes
2answers
5k views

Calculating license plate combinations

If there are $26$ available capital letters and $6$ available numbers $0$ to $9$ to complete a $6$ "digit" combination (a license plate for example) what is the formula for calculating the number of ...
2
votes
1answer
25 views

Invertible Uniform “PseudoRandom” Function

Perhaps this is better suited to a cryptography stack exchange, but I thought I'd try in mathematics in case this question is more obvious than I initially thought. I'm looking for a function ...
1
vote
1answer
15 views

Why does the average of a set of random numbers to the nth power approach 1/(n+1)?

I got bored and started running a Java program to mess with stuff like this. I did a boatload of trials and averaged them all together, first for a random number squared. Quick pseudo-code: ...
1
vote
1answer
30 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 ...
20
votes
8answers
3k views

3 random numbers to describe point on a sphere [duplicate]

I'm currently working on a problem involving computer graphics and got into a discussion about whatever or not constructing a 3d vector out of 3 random points uniformly distributed points between -1 ...
0
votes
0answers
44 views

Mathematical expression for randomly selecting from two integers

I wonder what is the shortest and correct mathematical expression for the below: $x =$ random$(1$ or $2)$ where there is equal probability of selecting 1 or 2. Thanks for your help.
-1
votes
0answers
23 views

Probability Density Function of Random Process

I have a signal s(t)=t for t is between 0 and 3. I have a uniform random variable A between 0 and 10. The random process is defined as s(t-A). What is the probability density function of the process?
2
votes
0answers
31 views

Limiting distribution of infinite sum of weighted bernoulli?

Let $p_n$ be some fixed pulse, for example $p_n =e^{-n^{2}}$ We have an infinite sum $y = \sum_{n=-\infty}^{\infty} a_n p_{-n}$ where $a_n$ are iid bernoulli random variables taking the values $+/- ...
1
vote
4answers
794 views

Easy way to generate random numbers?

How random this numbers look, 21081461046286104621816 Here the system I used to get them, first pick the seed, 13128 then add the value of the 2nd number to the 1st and write it between them ...
3
votes
3answers
65 views

Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”

I am doing some tests with strictly increasing integer sequences whose gaps between consecutive elements show a "pseudorandom" behavior, meaning "pseudorandom" that the gaps do not grow up ...
1
vote
0answers
37 views

Mean Value of a Random Process

Consider a random process $X(t) = Z(t)\sin(wt-Q)$. Here $Q$ is a random variable taking values $q$ in $[-\pi/2,\pi/2]$ with PDF given by $$p_1^Q(q) = \frac{\cos(q)}{2}$$ $Z(t)$ is some random ...
1
vote
0answers
34 views

Practical example of superiority of randomized algorithm

I'm looking for an example to show my students of an algorithm for which randomization of some kind leads to better performance on average. And I don't want that randomization to be of the Monte Carlo ...
0
votes
1answer
51 views

Apportioning $100\%$ across $24$ hours

Assume I have value $x=500$. I would like to apportion this value $500$ across $24$ hours time period and would like it to be apportioned randomly. So if $500$ is $100\%$ the $x\%$ generated at random ...
0
votes
0answers
19 views

Covariance matrix of random vector of vectors

I am a beginner in statistics and tried to research my question online without much success. Motivation: I am working on an undergraduate project in cosmology. My problem involves several ...
24
votes
4answers
13k views

Picking random points in the volume of sphere with uniform probability

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of ...
0
votes
0answers
15 views

Using injective functions to prove randomness

Suppose we have a random subset of $\mathbb N$. My intuitive notion of randomness represents the ability to encode arbitrary real number. Assume that $A$ is a random subset according to the above ...
1
vote
1answer
104 views

Mixture process conditioned on another mixture process

Let $N(T_1)$ is Poisson counting mixture process and $M(T_2)$ is another independent Poisson counting mixture process, such that $$ N(T_1) \sim (1-p) \cdot \operatorname{P}(\lambda_0T_1)+p \cdot ...
0
votes
0answers
3 views

How to calculate $E[P_e(Y)] = \int^\infty_{-\infty}Q(\sqrt {2y})f_Y(y)dy$

X is Gaussian (0,1) random variable with CDF $F_X(x)$ $$ Q(x) = 1-F_X(x) $$ Y is an exponential $1/\gamma$ random variable with PDF $$ f_Y(y) = \begin{cases} (1/\gamma)e^{y/\gamma} &y\ge0 ...
1
vote
2answers
37 views

Kth largest element from N chosen with a non-standard distribution

I have the following problem: $n$ values $U_1, \ldots, U_n$ are chosen randomly and independently from the interval $[0,1]$. When choosing $U_i$, the probability that $U_i$ is smaller than a ...
25
votes
9answers
14k views

What's the difference between stochastic and random?

What's the difference between stochastic and random? I've read in the Portuguese Wikipedia that there's a difference, but I still didn't see this point on English Wikipedia.
0
votes
2answers
42 views

How to calculate the average number of guesses made to guess a number between 0 and 31?

I know this question is very basic, but my question is more about Mathematical thinking. How to think mathematically: Suppose person $A$ chooses a number from numbers between $0$ and $31$, and ...