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

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1answer
88 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?
2
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1answer
26 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
87 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
60 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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0answers
23 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 ...
0
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1answer
38 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 ...
2
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1answer
19 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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0answers
31 views

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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0answers
14 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
53 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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0answers
24 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 ...
3
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2answers
92 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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0answers
27 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 ...
3
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1answer
24 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 ...
1
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2answers
30 views

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
16 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 ...
2
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1answer
35 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, ...
1
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1answer
19 views

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 ...
2
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0answers
32 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 ...
0
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1answer
26 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 ...
0
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1answer
24 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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0answers
47 views

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 $O(n)$ ...
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2answers
29 views

Inequality on probability of the sum of random variables

For Independent random variables $X_{i}$ can we write down the following inequality? $$\Pr \left\{ {{X_1} + {X_2} + ... + {X_n} \le k} \right\} \le \Pr \left\{ {{X_1} \le k} \right\}\Pr \left\{ {{X_2} ...
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1answer
39 views

Random process of the form $X(t)=A^t$ where $A$ is a given random variable

We’re going to look at a random process, which is a sequence of random process, which is a sequence of random variables that depend on time. Let $X(t)=A^t$ where $A$ has the density $f_A (a)=(3/8) ...
3
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1answer
33 views

Probability of random assignment to form pairs

So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ...
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0answers
13 views

Show equalities of random variables

In the text we showed that a geometrically distributed random variable W has the lack of memory property. Now assume that the range of W is {1,2,3...} and that P(W = j + 1|W>j} = p for j = ...
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0answers
37 views

random number generation from joint distribution where marginal are independents and known

Let $f(x,y)$ be the joint distribution of $X$ and $Y$, where $X$ and $Y$ are both positive and continuous. I can decompose the joint distribution as $f(x,y)=f(x|y) \, f(y)$ and I can easily generate ...
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0answers
205 views

Random 3D points uniformly distributed on an ellipse shaped window of a sphere

How can I generate random points uniformly distributed on the surface of a sphere such that a line that originates at the center of the sphere, and passes through one of the points, will intersect a ...
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2answers
59 views

Sum of Binomial Coefficient products

I am trying to prove that $$\sum\limits_{y=0}^d \frac{{2x \choose y} {2d-2x \choose d-y} }{2d \choose d} = x $$ So far, I have tried using induction on $d$ but I am having trouble using the ...
-1
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1answer
62 views

How can I use a Gamma Random Variable to Aproach the Expected value of a exponential random variable function?

I´m working on trying to approach the value of $E\left[ \dfrac{e^x}{x+1} \right]$. Where $x$ is an exponential random variable. All that data I have to work with is a gamma random variable with ...
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2answers
59 views

Solved - Use Random Number to Derive # based on Probability Table

Update I was able to derive the algorithm and implement it into excel. Thanks for the formula. Something like: ((z-xlbound)/d(x)*d(y))+ylbound See original sheet at end of post Original Post ...
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2answers
44 views

How to check if a sequence is random?

When I was thinking about various types of pseudo-randomness, the following question struck me: Suppose that a sequence $a_n \in \{0,1\}$ is given. Is there a way to check if it is genuinely ...
3
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0answers
68 views

Are humans capable of thinking of a series of random numbers? [closed]

I read in a book today ( the computer music tutorial by curtis roads), that humans are not capable of imagining any long series of truly random numbers. Apparently, the only way to generate a series ...
2
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0answers
22 views

What is the test of for randomness

How do you tell if a large sequential sample taken from a 0-1 rectangular parent - and presented as random - is truly random?
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1answer
34 views

Selecting a random orthogonal polygon

For a certain demo application, I want to create at random a rectilinear polygon with a given number of corners. Selecting random $x$ and $y$ coordinates of each corner is not a good method, since ...
1
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2answers
70 views

Expected Value of Identically distributed random variables

I have a very quick question regarding the expected value of two random variables $X,Y$ that are identically distributed and not necessarly independent. Is this equation valid? $E[XY]=E[X^2]$ If ...
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0answers
34 views

The probability that uniformly distributed integers sum to a given integer

A recent CTF had a problem involving the summation of randomly distributed integers. Specifically: Consider a set $\{X_m\}$ of $M$ integers uniformly selected (with replacement) from the set of ...
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2answers
23 views

Random number distribution from a different distribution

Suppose I have a random number generator that generates random numbers $x$ with a normal distribution $p(x) \propto e^{-x^2}$ (modulo normalization, but lets keep it simple). Now, out of these ...
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0answers
11 views

Is it possible to use multiple time scale algorithm here?

Suppose a random sequence is being generated (the next term generated depends on the previous term, but we don't know any distribution) until we hit some specific number. We want to calculate the ...
0
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1answer
21 views

Generate random results in a continuous field

How can we generate random results for a field like economical predictions where there is no limited number of results (contrary to a coin with 2 results) and also contrary to a random walk with steps ...
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1answer
77 views

How Many Unique Character String Can Be Made From 62 Characters?

I'm working on a programming algorithm and need a little math help. I'm in 10th grade and I think the question I'm asking is actually a permutation and combination logic question. Okay, so I've 62 ...
7
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5answers
692 views

Is the product of uniformly distributed numbers, uniformly distributed too?

My question is simple, I think. If we took two random natural numbers $a$ and $b$ uniformly distributed in a specific range $[c,d]$, is $ab$ a uniformly distributed too? What if $a$ and $b$ are not ...
2
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2answers
49 views

How to find out number of possible outcomes by trying over and over?

While working on my network exploration tool project, I've ran across the problem of reliably determining number of possible exit addresses of a tunnel with single entrance. I've came up with ...
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0answers
14 views

What it means by “asymptotic normality” properties of a random matrix?

I know that for the case of a random variable and a random vector, one can using (multivariate) density of normal distribution and concepts of convergence to define an asymptotic normality of a random ...
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0answers
42 views

Understanding Sobol sequences

Can someone explain to me in simple terms, how Sobol sequences work? The wikipedia article is fairly technical. They look pretty interesting. So I shall describe (whatever little I know) in short the ...
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0answers
14 views

Distribution of cut-off pseudo random numbers via linear congruential generator

I am currently using congruential prng as described here: Wikipedia. Now I need pseudo random numbers in the interval $[0, N), n < m$ where $m$ is the divisor of the modulus calculation. I want to ...
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0answers
39 views

Is the probability of variable independence = 0?

I understand the concept of independence to be dichotomous- events are either independent or dependent. And while there are infinitely many ways for events to be dependent and only one way to be ...
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2answers
59 views

What is the expected value of this sum (drawing slips from a bag)?

Suppose there are thirteen slips in a bag, labelled 1-13. If I draw a 10, 11, 12, or 13 then I stop adding to the sum and return the sum. If not, I add the number I draw to the current sum and place ...
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0answers
39 views

Dimensional Consistency in Grids used in Optimization

I am working on an optimization problem in the research I am doing and my partner and I have found that in order to quickly converge on a solution using a specific PSO (the firefly algorithm - it's ...
1
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1answer
21 views

Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...