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

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8
votes
2answers
858 views

The pseudoness of pseudorandom number generators

Is there a reasonable statistic test one can do to standard random number generators (say, one of those that come built in in Python libs) which shows they are not really random? (By reasonable I ...
14
votes
3answers
1k views

choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries?

choose a random number between 0 and 1 and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds 1. What's the expected ...
17
votes
7answers
3k views

Generate a random direction within a cone

I have a normalized 3D vector giving a direction and an angle that forms a cone around it, something like this: I'd like to generate a random, uniformly distributed normalized vector for a ...
28
votes
5answers
2k views

How to find a random axis or unit vector in 3D?

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction. But in 3D I ...
14
votes
4answers
6k 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 ...
2
votes
1answer
388 views

Are squares of independent random variables independent?

If X and Y are independent random variables both with the same mean (0) and variance, how about $X^2$ and $Y^2$? I tried calculating E($X^2Y^2$)-E($X^2$)E($Y^2$) but haven't been able to get anywhere. ...
9
votes
2answers
2k views

How can I pick a random point on the surface of a sphere with equal distribution?

I've got a random number generator that yields values between 0 and 1, and I'd like to use it to select a random point on the surface of a sphere where all points on the sphere are equally likely. ...
8
votes
3answers
473 views

How do we check Randomness? [duplicate]

Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can ...
7
votes
2answers
478 views

Probability that a sequence repeats itself

Given an infinite sequence $a_n$ of uniformly random integers $0$ to $9$, what is the probability there exist an integer $m$ such that the sequence $a_1$ to $a_m$ is equal to that from $a_{m+1}$ to ...
4
votes
3answers
374 views

What's the chance of an explicit series of integers in a limited random distribution?

Say I collect 40 perfectly random integers between 1 and 400. What's the chance that any integer is repeated consecutively six times in such a random draw? What I'm looking for is the chance of ...
-2
votes
2answers
235 views

Is first order moving average a Markov process?

Given first order moving average $$ x(n) = e(n) + ce(n-1) $$ where $e(n)$ is a sequence of Gaussian random variables with zero mean and unit variance which are independent of each other, and $c$ is ...
13
votes
4answers
559 views

How to efficiently generate five numbers that add to one?

I have access to a random number generator that generates numbers from 0 to 1. Using this, I want to find five random numbers that add up to 1. How can I do this using the smallest number of steps ...
1
vote
2answers
284 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 ...
5
votes
2answers
512 views

Summing (0,1) uniform random variables up to 1 [duplicate]

Possible Duplicate: choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries? So I'm reading a book about ...
4
votes
1answer
386 views

Generate a set of random numbers with a normal distribution

I am trying to generate a set of N random numbers where the set has a normal distribution. I'm currently using a brute force approach: Randomly select N numbers from a normal distribution. Check ...
4
votes
2answers
738 views

Connection to Normal distribution

I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$. I've used the law of total ...
3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
0
votes
2answers
48 views

Finding derivative of this integral function.

I need help on finding the derivative of this: $$g(x) = \int_1^{x^2} (x-t)\sin^2(t)dt$$ I thought about taking out x and having it as a constant but how?
0
votes
2answers
248 views

How to check that a sequence of numbers is random? [duplicate]

I have a sequence of numbers like 1,7,22,45,12,96,21,45,65,36,85,14,51,16,18,17,16....65... IS there any formula to check whether the sequence is random or not ? In my case odd numbers are ...
121
votes
24answers
10k views

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
18
votes
8answers
2k 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 ...
31
votes
3answers
6k views

Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
8
votes
3answers
2k views

uniform random point in triangle

Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. This paper (section 4.2) says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following ...
4
votes
2answers
5k views

Difference between logarithm of an expectation value and expectation value of a logarithm

Assuming I have a always positive random variable $X$, $X \in \mathbb{R}$, $X > 0$. Then I am now interested in the difference between the following two expectation values: $E \left[ \ln X ...
7
votes
2answers
310 views

Accessible Intro to Random Matrix Theory (RMT)

I read this fascinating article: http://www.newscientist.com/article/mg20627550.200-enter-the-matrix-the-deep-law-that-shapes-our-reality.html Unfortunately all the other papers I find googling are ...
3
votes
1answer
1k views

Expected value in collecting a set of coupons

There are $k$ types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type $i$ with probability $p_i$ s.t. $\sum\limits_ {i=1}^kp_i =1$. If $n$ ...
6
votes
2answers
608 views

How to generate REAL random numbers with some random and pseudo random

I'm doing (with Java) a very simple simulator (Queueing Systems..) that needs many random numbers (more than $10^5$). I know that Java Random class would give me all the random numbers I need, and ...
6
votes
3answers
8k views

Sum of independent Binomial random variables with different probabilities?

suppose I have independent random variables $X_i$ which are distributed binomially via $$X_i \sim \mathrm{Bin}(n_i, p_i)$$. Are there relatively simple formulae or at least bounds for the ...
5
votes
1answer
602 views

How can we generate pairs of correlated random numbers?

If I can generate normal random numbers in $N(0,1)$, how can I generate two dependent random numbers, $Y_1$ and $Y_2$ with means $\mu_1$, $\mu_2$ and $\sigma_1$, $\sigma_2$ and correlation coefficient ...
5
votes
1answer
901 views

How/why does this noise function work?

How/why does this noise function work? ...
4
votes
2answers
569 views

Period of linear congruential generator

How can you calculate the probability distribution of the period length of a linear congruential generator? That is $X_{n+1} = (aX_n + c) \bmod m$ where $a$ is chosen uniformly at random from ...
3
votes
1answer
66 views

How to pick N random points on the surface of a torus?

Lets say a torus is given by its major and minor radius. How to pick a set of N random points on its surface, with equal distribution? ("equal distribution" = for any chosen subsurface of the torus ...
3
votes
3answers
616 views

How to randomly construct a square full-ranked matrix with low determinant?

How to randomly construct a square (1000*1000) full-ranked matrix with low determinant? I have tried the following method, but it failed. In MATLAB, I just use: n=100; A=randi([0 1], n, n); while ...
3
votes
1answer
347 views

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
3
votes
2answers
1k views

Probability of Random number repeating

In the situation of having a high entropy random number generator, that generates numbers in the range of 0 and 2,147,000,000. If i have a list of 1,000,000 integer values, what are the chances that ...
2
votes
1answer
88 views

Probability of picked cards to be smaller than the largest picked card

I have an assignment for my algorithms module that requires us, amongst other things, to find the equations for the following question. Edit - Question Updated You have n cards with pairwise ...
2
votes
2answers
176 views

How to efficiently generate a sequence of random integers that adds up to 1?

I want to randomly generate a sequence of 10 integers in the range [-2, 2] that add up to 1. The distribution of the outcome isn't important. I just want the sequence to be random and the sum ...
1
vote
0answers
40 views

Why does a Gaussian process have a gradient whose determinant is Gaussian?

I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry. Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) ...
1
vote
1answer
101 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
4
votes
3answers
174 views

Is average of two random directions also a random direction?

Given two uniformly random directions on a hemisphere, n0 and n1, is the normalized sum of these vectors also a uniformly random direction on the same hemisphere?
3
votes
1answer
264 views

Number of samples to predict the next number in a pseudorandom number generator

Let: $$R_{n+1} = (mR_n + b) \bmod{a} $$ Assume we know the values of $R_1, R_2, \ldots, R_L $. What is the minimum value of $L$ (if it exists) such that we can determine $R_0, m, b$ and $a$?
3
votes
1answer
400 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 ...
3
votes
1answer
85 views

How many random samples needed to pick all elements of set?

If repeatedly picking a random element from a set, what is the expected number of times I'd have to pick before seeing all the elements of the set? Edit: when picking an element, it is simply counted ...
3
votes
1answer
400 views

How do I calculate the probability distribution of the percentage of a binary random variable?

I have an urn containing balls that are all either black or red. I'm interested in discovering the percentage of balls that are red. But I can only sample from the urn (without replacement), so the ...
2
votes
0answers
94 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
2
votes
0answers
71 views

Probability of ball ownership

$N+M$ people play a game of balls. Initially, N people hold N green balls (each person holds a ball), and M people hold no balls. Assume $M<N$. Then, M red balls are divided randomly - each ball ...
2
votes
2answers
194 views

Are there any random variables so that E[XY] exists, but E[X] or E[Y] doesn't?

Are there any random variables so that E[XY] exists, but E[X] or E[Y] doesn't?
1
vote
1answer
37 views

Generate random numbers with a modified PERT distribution

I want to generate random numbers based on the modified PERT distribution. The modified PERT distribution is a special case of the beta distribution and is defined as: $$f_X(x) = ...
1
vote
1answer
215 views

Simple random sample without replacement

I have a data file from which I wish to create a uniformly distributed simple random sample, without replacement. Will the following algorithm give me an unbiased result? ...
1
vote
0answers
78 views

The mathematics behind Sobol sequences

I am using Sobol sequence as random number generator in a computer program. Beyond just making the program work, I would like to learn the mathematics behind the Sobol sequence (and other ...