Tagged Questions

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

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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 ...
1
vote
1answer
36 views

Random points inside a convex polytope

Given a convex polytope, defined by set of vertices $P = \{\mathbf{x}^{(i)}\}_{i = 1}^n, x^{(i)} = (x^{(i)}_1, x^{(i)}_2, \dots, x^{(i)}_d): \operatorname{conv}(P) = P$. How to generate uniformely ...
2
votes
0answers
58 views

Randomness in a sequence

For a function $f$ consider a random sequence $a_{n+1}$ can be either $a_n+f(a_n)$ or $a_n-f(a_n)$ Given that the next term in the sequence is subtracting $f(a_n)$ from the previous term 50% of the ...
5
votes
2answers
1k views

What is golden ratio doing in this computer code?

In this file (related to random number generation), there is following line: private const int MSEED = 161803398; which reminds on golden ratio. How come ...
12
votes
2answers
1k views

Apparent Paradox in the Idea of Random Numbers

This question is a bit less than rigorous, but it's only because I don't know how to formulate it rigorously. Suppose there was some machine, or function, or whatever that could output a random ...
0
votes
0answers
19 views

Spherical Sampling of Projected Disk

Given a 2D solid disk centered at some 3D point $\vec{x}$ with radius $r$ and normal $\vec{N}$, I need to compute a random unit vector from the origin that hits the disk. The vector needs to be ...
0
votes
1answer
16 views

PRNG to generate a known set of values

Say I want the first N numbers generated by PRNG to be some list [n1, n2, n3, n1, nx...]. Is there any way to accomplish this (basically find the appropriate seed ...
3
votes
1answer
72 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 ...
1
vote
1answer
38 views

All combinations for a picking from a set of three colored balls

I can't find this exact question. The specific question I have is this: If you have a bag with one red, one blue, and one green ball, the chance of picking any particular one is 1/3. Simple enough. ...
1
vote
2answers
55 views

Exercise on Random Variables

I´m struggling with a random variable exercise of a book I´m reading. Anyone has an idea of how to approach this problem? Thanks in advance :)
2
votes
2answers
45 views

Select an element without uniform distribution with a uniform random without iterations

I have N elements (numbered from 0 to N-1) and I must choose one but without the same probability. For example, I need that the 0 must happen 50% of times, 1 with a 25%, 2 with a 12.5%, etc. I don't ...
0
votes
0answers
102 views

Probability Question involving Probability Mass Function/Random Variables

Problem: When a paging system transmits a message, the probability that it will be received correctly by the appropriate pager is p. To ensure that the message is correctly received at least once, the ...
0
votes
2answers
37 views

Calculate random integer inside a range of real numbers

$$F : \Bbb R \times \Bbb R \rightarrow \Bbb N $$ $$F(\text{minReal},\ \text{maxReal}) = \text{randomInt} \in \left[\text{minReal},\ \text{maxReal}\right] $$ Let $r \in [0, 1)$ be a random value. How ...
0
votes
0answers
24 views

Select machines to minimise latencies between them

I am working in an optimisation problem. I am still trying to model it and solve it. The problem is: There is a number of different types of virtual machines. Each type has different hourly cost ...
0
votes
2answers
44 views

Expected value: Product of integers in a range

What is the expected value of $f(x)^k = f(x) f(x) \cdots f(x)$ if $f(x)$ is a function, that returns a random integer between $0$ and $x - 1$ inclusive?
1
vote
1answer
69 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) = ...
3
votes
0answers
31 views

What is the most important test for a uniform random number generator?

What is the most important test for a uniform random number generator ? Is there a single most important test or a set? I am a using some analytically arrived at answers to probability problems and ...
1
vote
1answer
22 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
0
votes
2answers
48 views

Probability Random Variable question Need Help Please

You have a set of ten light bulbs - the lifetime of each of them being given by an exponential RV with mean 1000 hrs. Find the probability that.... (a) at least 7 of the bulbs function for 1500 or ...
2
votes
1answer
29 views

Why is the formula for generating Van der Corput sequences called an Inverse Radical Function?

The Van de Corput sequence can be generated using the following formula: $\phi_b(n) = \sum_{i=1}^N { a_j \over b^{i-1}}.$ where this can be defined as the "one-dimensional sequence defined by the ...
0
votes
0answers
25 views

mean square convergence vs almost sure convergence

I saw a few examples that show that almost sure convergence doesn't imply convergence in mean square. Can anyone find an example of a random series that converges in mean square but doesn't converge ...
0
votes
0answers
40 views

Proving properties of Random Graphs

I am asking the question on a slightly abstract level and it may depend on the specifics but it would be great to have related references or ideas. Consider the random graph model $G_{n,p}$ where its ...
0
votes
0answers
13 views

How to normalise logarithm depending on the size of the range of possible values

I'm attempting to write a function which can be used to generate weighted random numbers between a set range, the size of which can arbitrarily grow and shrink, depending on some modifying value to ...
3
votes
2answers
103 views

What is the probability of a specific sequence of 11 digits occurring in a random sequence of one billion digits?

This isn't homework, I'm actually (please don't ask me why) wondering how likely it is that any particular 11-digit telephone number will occur in the first billion digits of pi. My probability course ...
1
vote
0answers
43 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) ...
0
votes
0answers
44 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
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 ...
1
vote
0answers
28 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...
2
votes
0answers
58 views

“Alice and Bob” problem with matrices

Alice has a binary matrix $A$, and Bob has a binary matrix $B$. Both matrices are of size $n \times n$. Alice and Bob want to check if the matrices are equal except for exactly $1$ entry. Alice has to ...
1
vote
1answer
57 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
0
votes
1answer
54 views

Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $E[||x||_2]$, $x $~$ ...
1
vote
0answers
21 views

Ideal shrinking of discrete randomness source

I have a discrete randomness source that emits random integer numbers in range [0..N) with uniform distribution. I need to reduce this distribution, limiting it to ...
3
votes
1answer
75 views

What constitutes randomness?

To begin with, I am not a mathematician, so the question might sound dumb :) Anyway, I have been playing 2048 and thinking on randomness and some things just do not seem very straight to me, when ...
2
votes
0answers
42 views

Knuth shuffle : Is there a reciprocal to the factorial?

I have looked into the Knuth collection shuffle algorithm with pseudorandom number generators. They say that a PRNG with a seed state of $19937$ bits (like one of the Mersenne Twisters) can shuffle a ...
2
votes
1answer
89 views

Primitive polynomials in LFSRs

I need help proving the following theorem. I found it many books but on every single one it says that they omit the proof because it is in every good textbook. THM Let $c(x)$ be a connection ...
0
votes
2answers
53 views

How to tell if a set is cyclic

I've been reading a bit about pseudo-random number generators lately. My understanding is that they generate numbers that seem random for all practical purposes but they are actually just really big ...
0
votes
1answer
44 views

probabilities for obtaining 3 dice result

I am developing an app that needs a 3 dice result and there are two methods I can use: method a) create a matrix of 216 possible combinations and choose 1 randomly method b) or generate 3 ...
0
votes
0answers
13 views

Sampling a path in graph where each edge has a fitness

I'm writing an heuristic algorithm for the Travelling Salesman Problem. In one step, I have to generate a random path of length $n$ in a graph $G$. There is a real number in $[0, 1]$ associated to ...
1
vote
3answers
65 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
1
vote
1answer
55 views

How to place random points on an egg?

While I was having breakfast this morning, I thought: "I know how to put random points on a sphere or a torus, but how to place random points on an egg (with uniform probability with respect to any ...
1
vote
1answer
197 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
1
vote
3answers
143 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
3
votes
2answers
86 views

Alice and Bob card game

I came across this puzzle online in an online Princeton thingy (course?): Alice writes down two integers between 0 and 100 on two cards. Bob gets to select one of the two cards and see its value. ...
0
votes
0answers
21 views

Mathematically safe, open TAN generation and verification

For an upcoming Arduino project I need to generate, distribute and verify TANs (transaction authentication numbers). Let me explain (and assume for a second this is happening on a bigger computer). I ...
1
vote
0answers
37 views

Calculate expected value E(|x-y|^2)

I have two random variables (X and Y) that are uniformly distributed from 2.16 to 6.81 both. And I need to find E(|x-y|^2). Is this correct: ...
3
votes
4answers
150 views

Does this sum converge or diverge?

Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal ...
2
votes
2answers
35 views

Generate random symmetric positive-definite matrix

Is there a simple way to generate a random matrix that is symmetric and positive-definite? The symmetry seems like it could be achieved by generating a matrix $M$ with independent random entries and ...
1
vote
0answers
45 views

random nonsingular matrices using matlab

Does anybody know how to generate a random nonsingular matrices using matlab? I use sprand (m, n , dens, 1)function to specify the condition number to be 1 right now.But it is too slow.Is there any ...
1
vote
2answers
69 views

Median of i.i.d. uniform random variables on the interval $[-1/2, 1/2]$

Let $X_1, X_2, \ldots , X_{999}$ be independent and identically distributed random variables on the interval $[-1/2, 1/2]$. Let $X_{500}$ be the empirical median; that is, $X_{500} = X_k$ for some ...
1
vote
0answers
15 views

Raffle between different groups composed by different numbers

I've got this issue, I need to prepare a raffle between teams for a cars race. Cars are grouped by teams. Rounds are 1:1, composed by different manches until the cars are done. Total number of cars is ...