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

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2
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23 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. ...
0
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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
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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
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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 ...
1
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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: ...
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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 ...
0
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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
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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
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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 $T$.
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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 ...
0
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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 $10,000$ ...
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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 ...
1
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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 ...
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 ...
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 ...
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 ...
0
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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
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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
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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 ...
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: ...
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
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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
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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
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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
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 ...
1
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0answers
35 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
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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 ...
0
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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 ...
0
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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 ...
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 ...
0
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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 ...
0
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0answers
20 views

Complexity of a randomly-traveled N*M grid

I made an algorithm that compute a random-path across a N*M grid using backtracking. It start at [N / 2][0] and shall finish at [N / 2][M - 1]. Each iteration he choose a random direction (Left, ...
2
votes
2answers
96 views

Is there a way to mathematically prove my teaching assistant is wrong?

Our design of algorithms class requires all students to enroll in an online $AI$ competition, where each team has to come up with a bot. Before the final lockdown, each team is allowed to challenge ...
1
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1answer
58 views

PDF of maximum of two random variables

X and Y have joint PDF $$\ f_{X,Y}(x,y) =\begin{cases} 1/15 & 0\le x \le 5, & 0 \le y \le 3 \\ 0 & \text{otherwise.} \\ \end{cases} $$ Find the PDF of $W = ...
0
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0answers
28 views

How can you distinguish modular exponentiation from random?

Let $N$ be the product of two primes and let $P$ be the smallest prime larger than $N$. Let the algorithm $R(N,s)$ return $s^{1/P} \pmod{N}$. Let the algorithm $\widehat{R}(N,s)$ pick a ...
1
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2answers
43 views

What is $cov(XY,X)$

Suppose I have two real valued random variables $X$ and $Y$. Let $cov(\cdot)$ denote the covariance operator. What is $cov(XY,X)$? Is it zero if $cov(Y,X)=0$?
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0answers
19 views

How to find COV(max(a,b),max(c,d))?

We have normal random variables with nonzero mean, $$ X_i \sim N(\mu _{x_i},\sigma^2 _{x_i}) $$ $$ Y_i \sim N(\mu _{y_i},\sigma^2 _{y_i}) $$ $$ Z_i \sim N(\mu _{z_i},\sigma^2 _{z_i}) $$ They are ...
1
vote
5answers
236 views

If I ask $1000$ people to choose a random number between $0$ and $999$, what is the probability that no one will choose a specific number?

Imagine I asked $1000$ people to choose a number between $0$ and $999$ (both inclusive, the numbers are not biased, they will be completely random) and write that number down. Now, after that, pick a ...
1
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1answer
26 views

are all the variables independent of each other

Consider three random variables $X, Y$ and $Z$. If $X$ and $Y$ are independent, $Y$ and $Z$ are independent, then does that mean that $X$ and $Z$ are independent too?
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1answer
65 views

What is the probability that more passengers will turn up for a flight than the aircraft has the seating capacity for?

Daily Airlines flies every day from Amsterdam to London. The price for a ticket on this popular route is \$75. The aircraft has a capacity of 150 passengers. Demand for tickets is greater than ...
59
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3answers
1k views

Mathematicians shocked(?) to find pattern in prime numbers

There is an interesting recent article "Mathematicians shocked to find pattern in "random" prime numbers" in New Scientist. (Don't you love math titles in the popular press? Compare to the source ...
3
votes
2answers
85 views

Probability: Store opening time

Smith has a small booth where he sells lottery tickets. Customers arrive according to a Poisson process of rate $\lambda$= 1 per minute. He will close the shop on the 1st occasion that $a$ minutes ...
0
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0answers
14 views

Skewness and Kurtosis Excess, maximum allowed deviation for randomly generated numbers

I'm programming a random number generator which generates numbers according to a user-specified probability distribution (Bernoulli, exponential, uniform, pareto, etc). To verify the correctness of ...
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
1answer
16 views

Plotting a random but continuous line

Hopefully my question makes sense. I would like to generate a function which is continuous at all points. The use of the word "random" is somewhat of a misnomer, since it is not truly random. It must ...
1
vote
0answers
25 views

Delta Method: Estimate the Variance of $T$

Let $X = (X_1,\ldots,X_n)$ be a random sample, where $X_1 \sim \mathrm{Bern}(p)$. Let $\lambda = e^p$. Question: By law of large numbers, $T=e^{(\bar{X})}$ is a consistent estimator for $\lambda$, ...
1
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1answer
32 views

A random invertible matrix

I work on a project, for these project i need to generate a square random invertible matrix. I found out how to generate a square random matrix, still i want to be sure that this is an invertible ...
0
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1answer
25 views

significance of a test for (javascript) pseudo-random generator

I wrote a test case that tests if the javascript pseudo-random number generation algorithm of a given implementation has a given number of significant bits. You can see and test my code here: ...
2
votes
1answer
59 views

the greater of two random variables

For two independent normal random variables with non-zero mean $$ X \sim N(u_1,\sigma_1) , Y \sim N(u_2,\sigma_2) $$ If we have the condition, $$E(X^2 ) > E(Y^2)$$ is this condition always ...