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

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Mean time for the renewal process

The system is as below. Energy keeps coming at a node with a constant rate ρ. Node has files of size exponential(λ) to be transmitted (with fixed rate of transmission $r$, time for transmission would ...
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38 views

Presentation of 2 images in a random but counterbalanced way

Problem: For 18 trials randomly a ‘left’ labeled image or ‘right’ labeled image is shown. The first 9 trials should contain the opposite number of left images as the last 9 (a.k.a. counterbalance). ...
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1answer
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Probability and combining random variables

Adrian cycles to school with a mean time of 20 minutes and a standard deviation of 5 minutes. Pamela walks to school with a mean time of 30 minutes and a standard deviation of 2 minutes. They each ...
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+50

Arithmetic background of this RNG code

I am trying to figure out the mathematical background of the random number generation of an old video game. It does iterations where it updates a 33-bit state consisting of the variables z (32-bit) ...
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17 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
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7 views

Integration of a Random Process

Let {$X_t$} be a strictly stationary random process defined for all time $t$; in particular, the distribution (PDF) at any time is the same. Let $Y$ be a random variable defined by $Y = \int_0^5X_t ...
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Even dirstibution of a small set of random choices into a small set of buckets

Is there a way to evenly distribute randomly selected small set of items from a relatively larger set into to a small number of buckets using a hash function? For ex: Randomly select 20 numbers from ...
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1answer
10 views

relationship between two normally distributed variables

Say I have two normally distributed independent random variables (X1 and X2) with the same variance but different means. How would I calculate P(X1 > X2)?
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1answer
22 views

Select K numbers from N numbers fairly

I want to fairly select K numbers out of an array of N number. I know that this problem can be solved using Reservoir Sampling but I want to know if this approach is correct too? ...
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11 views

How to generate a random number according to a user defined distribution.

Assume we have a user defined distribution as following: $Q(d)=\intop_{d-\frac{1}{2}}^{d+\frac{1}{2}}G(t,\mu,\sigma)dt$ Where G is Gaussian distribution with mean $\mu$ and variance $\sigma^2$. How ...
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6 views

Stochastic Process with mean reverting property

Here I am seeking for a definition of what kind of stochastic processes are called mean reverting stochastic process. That is, what are the properties that a stochastic process should obey in order to ...
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22 views

What is the deciability of randomness?

Let f(n) be a function that returns 1 iff n consecutive ...
3
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0answers
23 views

Random Sampling of vectors on the Complex Unit Sphere

This is my first post in these forums. Working in Mathematica, I would like to generate a large number (10000) of randomly sampled vectors on the complex unit sphere in n dimensions. I am not ...
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14 views

Randomness of a Sample

While studying about Sampling I went through a question in Data Collection. QUESTION Suppose your population is numbered 000 to 299. Use the random digit sheet by taking consecutive three digits. ...
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1answer
18 views

Probability and random number [closed]

If F() generates 0 or 1 with probability 1/2 each, how can we generate 0-7 with equal probability?
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2answers
35 views

Picking 3 random books probability problem

So the question is: Suppose a bookcase holds 6 chemistry, 5 math, 3 physics, and 8 computer science texts. if 3 books are selected, find the probability that none of the math texts are selected. My ...
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1answer
27 views

Iterated uniform random variables

Define a sequence of r.v.'s {$X_n$}$_{n\ge 1}$ iteratively, such that $X_1\sim\text{Unif}(0,1]$ and $X_{n+1}\sim\text{Unif}(0,X_n]$. Could someone please explain why this is equivalent to: Let a ...
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3answers
103 views

What is the probability of getting the exact number of expected digits ($0-9$) in $10^6$ digits of $\pi$?

I noticed that at $1$ million digits of $\pi$, none of the digits has the "perfect" expected $100{,}000$ occurrences. My question is what is the probability (if the digits are truly random) of at ...
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1answer
182 views

Random digits of $\pi$ seem somewhat predictable! Why is this not generally believed? [closed]

I decided to test my theory that I can guess the next digit of $\pi$ with greater than $10$% ($1$ out of $10$) confidence based alone on knowledge of previous digits. I just used a simple spreadsheet ...
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1answer
17 views

Random number generation in MATLAB with condition

How can I generate random numbers in MATLAB with the condition that each new number has a minimum difference with all previously generated numbers for that matrix. So, X = rand(1,20) would generate 20 ...
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1answer
28 views

Flipping coin(s) to decide who pays the bill

I'm not from a mathematical background. I found this video on YouTube rather confusing. I know basics of probability from school. My question: If a single coin is flipped among 3 people say A, B and C ...
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1answer
29 views

Probability of coin flip betting

Imagine a situation where you and a friend both have 5 dollars, and you play him in a 50/50 coin flip "duel" where if it flips heads you receive a dollar from them otherwise you lose a dollar to the ...
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1answer
23 views

Binomial cumulative probability

Here is the question I need help on : Let $X$ be a binomial random variable with p = 0.5 and n = 100. Give $P(X \geq 60)$ rounded to two decimal places without using a calculator (by using ...
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3answers
119 views

What is in clear mathematical terms the definition for a sequence of integers, to be called *random*?

Sequences of integers might be ordered, totally ordered,... For all such attributes we find definitions in clear mathematical terms. (1) But what is in clear mathematical terms the established and ...
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Sampling from a random distribution with a margin of error

A survey of a population is taken using sampling. It is determined that 70% prefer option A and 30% prefer option B with a margin of error being 5%. Typically when simulating the process with the ...
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2answers
33 views

What is the mathematical notation for a random number in a certain range?

In school, I was taught to denote a random variable like so: $$R=\{_{0}^{1}$$ This only goes for a random choice between two integers, like a coin flip. How do I write a variable that is equal to a ...
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2answers
41 views

Is there some predictability in (pseudo-)randomness?

I was wondering if random numbers actually do have some predictability. For example, the digits of $ \pi$ are supposedly random. However, the frequency of some digits "lags" that of other digits. ...
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1answer
82 views

$\pi$ normal to the base $10$ [closed]

If $\pi$ is normal to base $10$, why would we expect to find a string of ten $0$'s in its decimal expansion?
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1answer
21 views

Can a collection of random processes be not random?

A friend and I were having a debate about randomness and at one point, I said that it was possible to have a collection of random processes which were not random when "put together." He disagreed. ...
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2answers
51 views

generate random number from exponential distribution with random right truncation

I need to draw a random number from an exponential distribution (rate $mu$) that is right-truncated with the truncation value coming from a gamma distribution (shape $k$, rate $lambda$). My naive ...
2
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1answer
57 views

Good ways to sample $n$ identical and dependent random variables

I'm wondering if there's a good way to talk about sampling identical but dependent random variables where it's also easy to see how the distribution evolves as we move from $n$ random variables to ...
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20 views

Probabilistic Method in language membership decision

$F(x,r)$ is a random algorithm to decide membership of input $x \in \{0,1\}^n$ to a language $L$. $r \in \{0,1\}^n$ is randomly selected bits. We know the if indeed $x \in L$, then the probability ...
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1answer
31 views

Expected value of a sum of random events

Suppose there's a market that has decided to award its most loyal customers. The market sells a certain type of breakfast cereals with a single token in each box. There are n different types of ...
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1answer
16 views

Can I use one float random number to generate two random numbers, one discrete, one continuous?

I need two random numbers. The first one, u, is discrete and takes 70% of the time the value 0 and 30% of the time the value 1. The second one, v, is continuous and takes values uniformly inside [0, ...
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Gaussian random processes

Let A B and C be Gaussian processes. If A&B are jointly Gaussian, and B&C are jointly Gaussian then A and C are jointly Gaussian? Is this statement true? and can it proven? (or any ...
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12 views

Will XORing any data with random data produce a random result?

Provided you have a stream of input data and a stream of random data both in the set (0,1). The random is data truly random, that is, unpredictable by the user and ...
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3answers
46 views

Generate random number with specific probability distribution

Well consider I have a uniform random number generator. How would I craft a function which takes as parameter this RNG, such that the distribution is following a given function? Or more to the point: ...
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18 views

Increasing the speed of a depth first search.

I am looking for ways to increase the speed of a depth first on a graph when trying to find a path from a source node to a destination node. One thing I have tried is to perform a bidirectional ...
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2answers
76 views

Probability a product of $n$ randomly chosen numbers from 1-9 is divisible by 10.

I'm working on a problem where each number is chosen randomly from 1-9. Given $n$ numbers chosen in this manner, we multiply all of these together. I'm looking for the probability that this product is ...
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25 views

Divergence of series

Given that a is a non 0 real number and ck is a series of real number for all k natural number where $\sum_{k= 1}^{\infty} c_k^2 = \infty $. If $c_k \to 0$, proof that $lim _{n \to ...
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1answer
20 views

Distribution of numbers $n = \max(x, y)$ where $x, y$ are random numbers between $0$ and $1$

I define a function $$f = \max(\mbox{rand}(0, 1), \mbox{rand}(0, 1))$$ such that $f$ returns the maximum (greater number) of two random selected numbers between $0$ and $1$. Plotting a histogram for ...
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2answers
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Variance of sample mean (problems with proof)

Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central ...
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0answers
15 views

How do you count a grouping without a sequential count?

So, rather than explaining how this problem pertains to the actual situation I think its easier and a great deal less work to give you a situation that you can visualize. Imagine a person who has ...
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1answer
29 views

Analogous of Markov's inequality for the lower bound

Consider a positive random variable $X$ and call $E[X]$ its expectation. For any positive $a \in \mathbb{R}$, an upper bound for the probability of $P(X>a)$ is provided by the Markov's Inequality, ...
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1answer
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Random conversions

I came across a question in StackOverflow which states the following, all is based on natural numbers: Given the function rand5 (which produces random natural numbers 0-4), use it to generate a rand7 ...
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0answers
29 views

How do you calculate randomness?

Suppose I receive a list of 1 million coinflips, and I want to know how likely it is that the list was randomly generated. My first thought would be to count the number of heads and tails, which ...
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0answers
34 views

Derivative of stochastic process

I have a set of data of a random process (one sample path). The process is sampled every 10 min and each sample is a 10 min average from a sensor. I can compute the statistics of the random process, ...
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1answer
21 views

Is there a name for the following random process?

I have a random process which seems to oscillate between extremes in terms of trending but which is locally like a Brownian motion or a fractional Brownian motion. Is there a name for such a ...
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1answer
22 views

What is the meaning $P[\frac{1}{n}\sum_{k=1}^{n}Z_k \le \frac{1}{2}\text{ for infinitely many }n]=0$

Let $Z_1, Z_2,\ldots$ be independent identically distributed (i.i.d) binary variables with $P[Z_i = 1] = 1-\alpha $ for some $\alpha > \frac{1}{2}$. Using the transformation $X_i=2Z_i-1$ together ...
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Generating a random combination in O(k)?

I need to generate a "fair" random combination of $k$ items chosen from $n$ choices. All the algorithms I've been able to find so far (reservoir sampling, Fisher-Yates shuffle, ...) are of $O(n)$ ...