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

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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) ...
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0answers
51 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 ...
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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 ...
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0answers
45 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 ...
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0answers
65 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 ...
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1answer
92 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} \\ ...
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1answer
387 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 $~$ ...
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0answers
82 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 ...
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1answer
97 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 ...
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0answers
57 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
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1answer
142 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 ...
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2answers
63 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 ...
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1answer
45 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 ...
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0answers
17 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 ...
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3answers
87 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 ...
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1answer
74 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 ...
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1answer
482 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 ...
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3answers
1k 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. ...
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2answers
160 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. ...
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0answers
27 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 ...
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0answers
40 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: ...
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4answers
173 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
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2answers
51 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 ...
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0answers
216 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 ...
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2answers
173 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 ...
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0answers
18 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 ...
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2answers
68 views

A binomial random number generating algorithm that works when $ n \times p $ is very small

I need to generate binomial random numbers: For example, consider binomial random numbers. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads ...
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1answer
54 views

Random processes

I hope someone could tell me how to explain that "random process is continuous by probability" and "random process is differentiated by probability"? I know that definitions are these: Given a time ...
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1answer
223 views

Finding the expected value of the length of a minimal spanning tree of n randomly generated nodes bound in a box with edge length a.

Say we specify a number (n) of random points (x,y), bound within the axes and x=a, y=a. Given the number of points and the constraints on the boundaries, how can you calculate the expected value of ...
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2answers
104 views

Calculating the expected value of n randomly generated numbers?

Say I have a random number generator that will generate x numbers - not necessarily integers - on the continuous range between a and b. How can I calculate the expected values for these numbers? My ...
2
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2answers
859 views

Simple algorithm for generating Poisson distribution

I found a very simple algorithm that draws values from a Poisson distribution from this project. The algorithm's code in Java is: ...
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0answers
32 views

Random noise and i.i.d's

In a book it is written that the quantity $\epsilon(\theta)$, called random noise, can be assumed to be i.i.d's when it does not depend on $\theta$. When it depends on $\theta$ it is said that the ...
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1answer
32 views

Am i in the right direction on this probability/random distribution question?

To improve the operation in the control tower of an airport, air traffic control engineers are assessing the delay due to taxi-out time, which is the duration between pushback and takeoff. suppose ...
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2answers
166 views

How to find binomial pmf with probability = another random variable

Assume that $Q$ is a random variable with density proportional to $q$ for $0 < q < 1$. Given $Q = q$, $N$ has a binomial distribution with parameters $n$ and $q$. What is the probability mass ...
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1answer
156 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 ...
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3answers
60 views

Random matrices in coordinate independent way

How to generate a random matrix in a basis independent way (so that the random distribution does not change if the coordinates are rotated)? I am especially interested in generating random rotation ...
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0answers
36 views

Probability of a specified sequence in a random data set

This is a problem which I have encountered while programming, but I imagine this community would be better able to solve it. Suppose we have a number, N, of boxes in a row. In each of these boxes is ...
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1answer
41 views

probility, placing balls, covariance

Can you please help to see where I did wrong? There are 10 balls, and each ball to be place in bin 1 and bin 2. Each ball is placed indepedently. Let X be the number of balls in bin 1 and Y be the ...
2
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1answer
51 views

probability, transformation on Random Variable

This is a more general question about the transformation of a random variable. Say X is given as a certain distribution, and Y=g(X). If it asks to compute the pdf of Y, I am having trouble to ...
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4answers
58 views

The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
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2answers
96 views

Dice-Game with two-twenty sided dice.

EDIT: I'll give this another try, trying to be clearer. The game is played like this: Player A roles two-twenty-sided dice and multiplies the two integers together to get some integer, say x, with $ ...
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1answer
28 views

probability, depedence => uncorrelated?

One quick question. Two random variables are dependent, so it must be uncorrelated? Also for the terminalogy, does the term "uncorrelated" means two random variables have a corvariance factor of 0? ...
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1answer
26 views

probability, please help on bayes question

I dont need exact answer, I just need help to judge wether my following method is correct or not. Question:A physician has 5 patients. There are treatments A and B. Physician gives treament A to 3 ...
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1answer
25 views

probability, at least one is not present

you have $5$ red balls, $10$ green balls, and $15$ yellow balls in a balls. You randomly choose $5$ without replacement. What is the probability that at least one of the $3$ colors is not present ...
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1answer
41 views

probability, need help on the marginal densities

I need help on the marginal densities. In particular, I know you just integrate the joint pdf f(x,y) from y=-infinity to +infinity, but in the context of the below question, I have trouble to define ...
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1answer
95 views

probabilty, random variable independent

Let $X$ and $Y$ be independent Poisson random random variables with ($\lambda=1$). Are $X-Y$ and $X+Y$ independent? Justify My attempt: $X-Y$ => random variable is $0$. $X+Y$=> Poisson of ...
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2answers
55 views

probability, indicator random variable

Let $A,B,C$ be independent events with $P(A)=P(B)=P(C)=\dfrac{1}{2}$. Let $X$ be the indicator r.v. of the event $A \cup B$ and $Y$ the indicator r.v. of the event $B \cup C$. Compute ${\bf E}[XY]$. ...
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1answer
124 views

Generate some random matrix with given rank

Very often for creating new exercises (I teach basic matrix algebra), I need to a find a matrix $A$ such that: it has integer coefficients, not too big (in order to avoid big numbers computations) ...
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2answers
58 views

probability: arithmetic with random variables.

I have a question of using arithmetic on random variables. Please refer to the following question, to which I will present my solution using the arithmetic (which I thought it's correct but actually ...
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1answer
149 views

Random points in sphere with probability p(r)

How to pick random uniformly distributed points in a sphere has been asked before. The difference is that I don't want uniform distribution, rather I would like the number density to scale by ...