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

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48 views

Selecting random $k$ elements from set of $n$ [on hold]

I am looking for way to select to $k$ items from set of $n$ items where $n$ is of order of $10^9$ an $k$ is of order of $10^2$. Probability of selecting each element should be equal. What algorithm ...
3
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1answer
24 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
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1answer
33 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
51 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
46 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
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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
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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 ...
0
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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
28 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 $+/- ...
0
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0answers
29 views

Probability density function with two peaks and skewness

I have plotted a probability density function on a graph. With one line from $(0,1)$ to $(1,0)$ and the second line from $(1,0)$ to $(2,1)$. The area under the lines sum up to $1$ and all values of ...
1
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0answers
36 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
42 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
33 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
16 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
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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
votes
2answers
41 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
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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
56 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
27 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
41 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
18 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
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5answers
227 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?
0
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1answer
63 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
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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
102 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
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0answers
24 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
31 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
22 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 ...
0
votes
0answers
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
votes
1answer
158 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 ...
0
votes
2answers
189 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 ...
1
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1answer
52 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?
4
votes
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 ...
1
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0answers
28 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? ...
1
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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 ...
0
votes
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 ...
0
votes
1answer
16 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 ...
1
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0answers
42 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 $X$, ...
-1
votes
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 ...
0
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0answers
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: ...