Questions relating to (pseudo)randomness, random oracles, and stochastic processes.
5
votes
2answers
247 views
Name this paradox about most common first digits in numbers
I remember hearing about a paradox (not a real paradox, more of a surprising oddity) about frequency of the first digit in a random number being most likely 1, second most likely 2, etc. This was for ...
0
votes
2answers
157 views
Random and Pseudo-random number generation
I heard that computation results can be very sensitive to choice of random number generator.
I wonder whether it is relevant to program own Mersenne-Twister or other pseudo-random routines to get a ...
1
vote
1answer
197 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 ...
1
vote
1answer
181 views
Generate a random variable by composition
If I want to generate a random variable with such a pdf: $f(x) = \frac{5}{12}[1+(x-1)^4], (0 \leq x \leq 2)$ by composition, what should I do?
Some of my thought is that I can first make it into ...
4
votes
3answers
2k views
What's the difference between stochastic and random?
What's the difference between stochastic and random? I've read in the portuguese wikipedia that there's a difference, but I still didn't see this point on english wikipedia.
3
votes
2answers
1k 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.
...
1
vote
1answer
80 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 ...
9
votes
3answers
444 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 ...
1
vote
1answer
145 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 ...
2
votes
2answers
212 views
Probability Theory - random number hit from a pool
With the given data:
After picking 30 random natural integers in a pool of X-natural-numbers (numbers do not disappear from the pool after picking). The probability of NOT picking 1 pre-defined ...
1
vote
1answer
90 views
pseudo-random permutation of $[0,N)$
Given a positive integer:
$$\begin{align*}
N \in \mathbb{Z}^+
\end{align*}$$
I would like a function:
$$\begin{align*}
f : \mathbb{Z}^2 \rightarrow \mathbb{Z}
\end{align*}$$
such that
...
1
vote
0answers
122 views
Non-uniform random walk
I'm searching a solution for this problem:
Given a segment of length $L$, from $0$ to $L$ divided in $N$ subsegments of the same length, a particle, starting from the subsegment in $x_k$ has a ...
5
votes
1answer
58 views
Interpretation of a simple probabilistic term in a calculation
I'm reading through my notes on the evolution of random graphs and have come unstuck trying to figure out the meaning of a probabilistic term which appears, and was hoping you could help - it's not ...
0
votes
1answer
245 views
Why are random numbers necessary for a Monte Carlo simulation?
This may be somewhat of a question with an obvious answer, but I can not seem to understand the necessity of "truly" random numbers to make a Monte Carlo simulation a good one.
I understand that not ...
2
votes
2answers
100 views
Need Randomness within constraints
The wife wants to make a quilt from squares of fabric cut from my old shirt. Awesome.
Situation:
2” x 4.5” blocks of fabric containing various patterns.
Quilt should be 17 across (4.5” the long ...
0
votes
0answers
45 views
Sequence of r.v.'s convergence
If we have to study almost sure convergence, or other kind of convergence, how to chose the random variable to which the sequence of r.v tends if it is not indicated?
1
vote
2answers
270 views
Probability, Joint Distributions, Standard Normal
I'm working through a course in Probability (2nd/3rd year) and would like to clarify some idea on joint distributions.
Suppose for example we have independent random variables $(Z_1,Z_2)$ from a ...
1
vote
0answers
62 views
Bayesian Network: Probability distribution of random variable itself a random variable
I'm doing a thing with a Bayesian Network. There is a tool to analyze such networks and there is a "doubt" setting in [0, 1]. If the certainity of a prediction is less than that value, then it is ...
2
votes
3answers
184 views
Validating results of a Monte Carlo integration
Suppose I want to use the Monte Carlo integration method to compute the following integral
$\int_D (e^{x^{2}} + e^{y^{2}}) \; dx \; dy$ where $D$ is some regular hexagon. I have managed to write the ...
0
votes
0answers
23 views
Evaluating randomized experiment with false negatives
I'm trying to evaluate a randomized experiment which produces positive or negative results, given an input value. I'm interested in finding all values for which the result is negative, but I get quite ...
0
votes
2answers
74 views
Convergence of sum of independent random values
If $f(x)>0$ is any function st. $\sum_{j=1}^{\infty}f(j)=\infty$. And $a_n=f(n)$ with probability 50%, else 0.
Does $\lim_{n\rightarrow\infty}((1/\sum_{j=1}^nf(j)) \sum_{k=1}^n a_k)=1/2$ almost ...
0
votes
1answer
54 views
How to scale a random based on a function?
Say I have a function function $f(x)$ returning any real between 0 and 1, for a $x$ between 0 and 1.
I want to get $n$ randomly generated values of $x$, based the probablity they occur from $f(x)$.
...
1
vote
1answer
369 views
How does one interpret the meaning of a stochastic derivative?
My understanding of derivatives is in the difference quotient limit sense... How does one interpret the meaning of a stochastic derivative? How can one possibly differentiate a random variable? What ...
2
votes
2answers
129 views
converting random sequence
There is stream (sequence) of uniform random integers $x_i$, each integer in $[0,N-1]$ range, where N is not power of 2.
I need to convert it to sequence of integers [0..255] (bytes), such that ...
1
vote
1answer
69 views
Finding a subsequence (of a very long sequence) which does not sum to an even number
[Edited. I've revised to problem to focus on the special case of the integers modulo 2.]
You are given a function f from binary strings x ∈ {0,1}n to the integers, or (without loss of ...
5
votes
1answer
311 views
Generating geometrically-distributed variates over an interval given endpoints and expected value
I want to generate random variates from a truncated geometric distribution over the interval $[0, n)$ with specified expected value, $0 \le E < n$. The obvious way to do this seems to be to sample ...
2
votes
0answers
73 views
Modeling Sample Covariance Matrix based on concepts from Random Matrix Theory
I am working on a signal processing problem where I want to model the measurement sample covariance matrix (SCM) as random matrix and hence use the results from Random Matrix Theory (RMT).
Let ...
0
votes
0answers
132 views
Uniform Random Points on a triangle using only edge plane normals
For a triangle $ABC$ in 3D (each point has x, y, z coordinates) is it possible to generate uniform random points on the triangle from only the following data:
Normal of the triangle plane $N = ...
4
votes
2answers
193 views
What type of distribution would rand()/rand() produce?
If rand() is a function which produces a linearly distributed random number over a range not containing zero, then what type of distribution would rand() / rand() produce?
I know it would center at ...
1
vote
0answers
576 views
Generate random number from Gaussian, Cauchy and Levy distribution [closed]
I am working on Genetic Algorithm. I have to generate random number from above three distributions. How can I do this?
1
vote
3answers
133 views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
1
vote
1answer
52 views
How to prove that the expression E(z'Az) (for a random vector z) is …
I am looking to prove the following
Let $z$ be an $m\times$ 1 random vector with $E(z)=\mu$ and $\operatorname{Cov}(z)=V$ and let $A$
be an $m\times m$ non-stochastic matrix. Then the following ...
0
votes
3answers
64 views
how much the entropy change going from 6 digit, to 8 digits with several sequence restrictions?
One system I use just changed their password policy. Previously, the only requirement was 6 digits. (e.g. 123456, 111111)
Now it's 8 digit, with no bigger then 3 digit sequence, and no pattern such ...
1
vote
1answer
94 views
Period of least significant bits of linear congruential random number generator
On a practice exam for a course on stochastic simulations I encountered the following question:
Show that the least significant $n$ bits must repeat with a period $2^n$
for a congruential random ...
1
vote
3answers
223 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
121 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 ($\bmod$). ...
6
votes
3answers
2k 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
2answers
60 views
Expectation of random callers distribution
In a country with $N$ people, if everyone calls one random person in the country, what is the expected number of people who dont receive a call?
If $f(m,n,a,b)=$Expected fraction of people who ...
1
vote
0answers
91 views
Correlated Random Numbers Between A Range
I am trying to generate random numbers within a range say 57 to 107 which are correlated by a factor of 0.8. The numbers are for illustration simplicity only.
2
votes
2answers
159 views
Are probabilities proportional to the distance traveled in a random walk? What if the initial position is a bit biased?
A marker is placed at zero on the number line and a fair coin is flipped. On each flip we move one unit to the right. If it lands on heads, the marker is moved one unit up. If it lands on tails, the ...
-1
votes
1answer
290 views
runs above and below the mean test
In the runs above and below the mean test of independence, the mean is
MEAN = (2*n1*n2)/N - (1/2)
So here, are n1 and n2 runs above and below or ...
6
votes
1answer
167 views
If W is a random matrix with variance $\mathbb{E}[W W^{T}]$, what's $\mathbb{E}[W^{T} P W]$?
I know quite a few identities about quadratic forms of random vectors, but I'm having difficulty coaxing something out of this quadratic form of random matrices. Suppose I know $\mathbb{E}[W W^{T}]$ ...
2
votes
1answer
63 views
random variable transformation
I'm having trouble with the following random variable transformation:
$Y = X^2 + X$
I am looking for the pdf of Y. I tried the following method:
$p_Y(y) = \int_{X} p_{Y|X=x}(y)\cdot p_{X}(x)dx$
and ...
3
votes
4answers
316 views
Random variables: How would you explain it to a beginner?
Different types of random variables:
(discrete) Binomial, hypergeometric, geometric, Poisson
(continuous) Uniform, normal, exponential
Random variables are very useful tools when solving simple and ...
3
votes
0answers
162 views
Simulation and Stochastic Processes
Supposing we want to take a sample from the distribution $p(x)=cp^*(x)$
where $c$ is the normalization constant and $p^*(x)$ is given by
$$p^*(x)=0.5\exp(-(x-\mu_1)^2)+0.5\exp(-(x-\mu_2)^2).$$
...
3
votes
1answer
141 views
Conditional Expectation and the floor function
I have a piece of code the produces random integers.
...
3
votes
2answers
97 views
Existence of a normal computable infinite pseudorandom sequence
Is there any computable infinite pseudorandom sequence of 0's and 1's which have been proven to be normal?
0
votes
1answer
148 views
Algorithms and Simulation
Supposing we want to take a sample from a $N(0,1)$ distribution and i can take a sample from a $N(0,σ^2)$.
(a) Construct a disposal/rejection algorithm with function $N(0,σ^2)$, which generates a ...
1
vote
3answers
438 views
Generate a set of random numbers with an average evenly distributed between two given values
1) I generate 1000 random numbers between 0 and 10 and take the average.
If I do the above action "many" times the resulting average values will be a normal distribution over 0 to 10. Correct?
What ...
0
votes
1answer
69 views
Test for randomness
I'm trying to write a program to compute a metric for the entropy in files to determine a probability that the file is compressed or encrypted. Compressed and encrypted files have very, very, very ...
