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

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8
votes
2answers
732 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 ...
16
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 ...
11
votes
3answers
5k 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 ...
25
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 ...
8
votes
3answers
418 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
409 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 ...
2
votes
1answer
318 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. ...
-2
votes
2answers
192 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 ...
11
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 ...
12
votes
4answers
538 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 ...
4
votes
3answers
341 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 ...
1
vote
2answers
240 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 ...
4
votes
1answer
348 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 ...
0
votes
2answers
44 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
230 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 ...
7
votes
3answers
1k 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 ...
5
votes
1answer
543 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 ...
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
556 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 ...
5
votes
1answer
810 views

How/why does this noise function work?

How/why does this noise function work? ...
4
votes
2answers
485 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
3answers
490 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
307 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
948 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
83 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
162 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 ...
4
votes
3answers
166 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?
4
votes
2answers
602 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
179 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
260 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
75 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
388 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
89 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
68 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
190 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
84 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 ...
1
vote
1answer
195 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
124 views

If I have a set of numbers, how can I prove that they are random?

If someone gives me a list of numbers and says they are entirely and completely random, how I can I verify this? EDIT: Let's suppose that a well-known string theorist told me he can produce a list ...
1
vote
2answers
102 views

Why does multiplying n by a random number give you a random number between 0 and n?

I have been following the tutorial at Codecademy I see that, as part of the process of creating a virtual die we multiply 'a random number between 0 and 1' by 6 to give us a random number between 0 ...
1
vote
1answer
138 views

How to find a random number between $x$ and $y$

Given some random number function rand, and some numbers $x$ and $y$, how do you find a random number $r$ such that $r\geq x \wedge r \leq y$? Previously i've ...
1
vote
1answer
124 views

Multiple instances of random generator

I wonder whether following methodology is correct. I need to introduce concurrent processing to my simulating application.I want to use Mersenne Twister random generator. I wonder whether I can ...
1
vote
1answer
165 views

Help with a short paper - cumulative binomial probability estimates

I was hoping someone could help me with a brief statement I can't understand in a book. The problem I have is with the final line of the following section of Lemma 2.2 (on the second page): Since ...
1
vote
3answers
269 views

Get random $x, y, z$ point inside a sphere?

What is the easiest way to get a random Point3D inside the sphere where is red?
1
vote
1answer
199 views

What does Linear Congruential mean?

How does one interpret the terms "Linear" and "Congruential" as in a "Linear congruential RNG"? I am used to linearity by $f(ax)=af(x)$. This does not seem to me to hold true in this case ($\bmod$). ...
1
vote
2answers
241 views

Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't?

Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't?
0
votes
2answers
42 views

Consider throwing two six-sided dice.

Let X be the sum of the two values and let Y be the product of the two values. What is the value of P(Y = i) for i = 1,2,3...36. I am having trouble approaching this problem. We are learning about ...
0
votes
1answer
51 views

how to generate Normally distributed random number?

I am looking for a function that can generate Normally distributed random numbers. I came to know about bux-muller transform but I didn't understood it completely what it is doing. Thus it would be ...
0
votes
1answer
80 views

Generating a sequence of $n$-bit random-like numbers that will never repeat.

I am actually aiming for n $\approx 100$ but obviously I could interleave the bits of two never-repeating $50$-bit sequences etc. I need to be able to generate the $n^{th}$ number quickly enough to ...
0
votes
1answer
237 views

maximum of two non central chi squared random variable

Let $$s_i \sim \chi(k_i, \lambda_i), i\in \{ 1, 2\}$$ be two non-central chi-squared random variables with $k_i$ degrees of freedom and $\lambda_i$ parameter of non-centrality I am wondering if ...