Questions tagged [random-functions]
Functions of random variables.
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1answer
42 views
What simple function $f(i)$ produces an evenly distributed pseudorandom output for $i \in [1, 2, …)$?
I'm looking for a transformation $N \to S$ where $N$ is natural numbers sequence, whereas $S$ is an infinite pseudorandom sequence that doesn't end up with a repeating pattern and has a uniform ...
0
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0answers
7 views
Can't understand finding characteristics of random variables
I have next random function as example in book: $X(t)=U*sh(t)-3e^{-3t}*V+t^2$
U and V - uncorrelated random variables. $U \in R(-3,3), V \in P(1,2)$.
Next step is to find the characteristics of ...
0
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1answer
35 views
summation of values of a uniform random variable
Say we have a random variable Y belonging to {-1,1}. Each time an ideal random number code-simulation generates a value for Y, using "Uniform distribution", let us give that value a symbol yi. So, ...
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0answers
22 views
Random recursive function that stays near the initial value
I am working on a procedural CG scene and have populated a starry sky with particles of random size. My goal is to make the stars twinkle - in this case, by varying their size and/or alpha channels. ...
0
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0answers
19 views
Example of higher random vector moments
While reading about random vectors, I learned that...
$$
E\left[\vec{X}\right] = \left[\begin{array}{cccc}
E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\...
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34 views
Would sampling the decimal digits of $\pi$ generate a white noise signal?
Discrete r.v. $X = \pi(d)$ (defined in another q of mine).
Discrete r.v. $Y = X - 4.5$.
q1: Would it be incorrect to deduce $Y\sim U(-4.5,4.5)$ from $X\sim U(0,9)$?
q2: If you answered no to q1, ...
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1answer
28 views
random experiment with two different functions on unit interval
Let $X=[0,1]$, and functions $f(x)=x$, $g(x)=2x$ mod $1$, and the probability of chosing $f,g$: $\mu(f)=\mu(g)=\frac{1}{2}$. Now if $x$ is the starting point, then what will be a general expression of ...
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15 views
Trying to build a random number generator with a non-linear probability, but what would the function look like?
It's my first post, so hopefully I don't muck it up.
I am trying to build a random number generator with a certain property that I'm having trouble implementing, describing, and searching for.
It ...
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0answers
16 views
An “Edgeworth Series-esque” approximation of ratio distribution using Monte Carlo methods. What is this method called?
I am hoping someone can provide me with the name of the following technique that appears to estimate the density of the ratio of independent random variables (although it could work for other ...
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16 views
Upper bound on the error when approximating expectation of a function of a random variable using Taylor series.
If I have a non-negative random variable $X$ whose distribution is not known but moments are known and I want to find the expected value of a function of this random variable, say, $f(X) = (1/(1+X)) + ...
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30 views
Probability of a inverse exp random variable larger than sum of inverse exp random variables
This is problem is an intermediate step that I encounter in my theoretical computer science study, to be specific, precision sampling.
Say we have a bunch of independent random variables $u_i \sim \...
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1answer
102 views
Solving second order nonhomogeneous ODE where the RHS is a random process
Context: I'm trying to characterize the metastability behavior of a digital latch.
I'm modeling two cross-coupled inverters as RC circuits with negative gain. One of the inverters has a source of ...
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0answers
48 views
Sample instances of random process given all temporal correlation functions?
I asked this question on signal processing stack exchange (question) but I wonder the general answer I am seeking makes the question better suited here.
Suppose I have a complex valued random process ...
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0answers
88 views
How to take ensemble average of a given function?
I am going through the calculation by Rawson et al. [J. Opt. Soc. AM. Vol. 70, No. 8, August 1980] and ran into seemingly simple issue with the derivation. I wanted to get some help on solving the ...
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1answer
31 views
A random variable in a denominator
For some random variable $X$ & $Y$, I need to calculate the pdf of $Z=\frac{X}{Y}$. I've managed to calculate $$F_Z(t)=\mathbb{P}\left(\frac{X}{Y}\leq t\right)=\mathbb{P}(X\leq t\cdot Y)\,,$$ but ...
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16 views
An example of random functions that are stochastically equivalent but aren't modifications of each other.
I'm trying to show that stochastic equivalence of random functions (in the broad sense) doesn't imply that they are modifications of each other. For this, I'm looking for a counterexample. Perhaps ...
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1answer
66 views
Can a hypercomputer solve random sequences? [closed]
I would love to know the answer to this question. Lets have a hypercomputer which is capable of doing an uncountably many computational steps in finite time with infinite memory. Now could such a ...
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7 views
An example of non measurable random function with measurable trajectories.
I'm looking for an example of non measurable random function with measurable trajectories. I think here I need an indicator of some unmeasurable set. But I still don’t understand exactly which ...
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0answers
35 views
Expected value of a random process which is a function of a Wiener process
I do not quite know how to attack and proceed with the following problem:
Let W(t) be a Wiener process. In that case, calculate the expected value of random process Y(t)=W^2(t)cos^4(W(t))exp(−3W(t)/2)...
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0answers
39 views
Expected value of random function
For $k \in \mathbb{Z}_+$ and a given $A \in \mathbb{Z}_+$, I know that
$\mathbb{E}\left[\widehat{f_A}\right] \leq A - \frac{A^2}{k}$, where $\widehat{f_A}$ is a random variable dependent on $A$.
Now, ...
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2answers
47 views
Differentiate a function of random variables
Suppose I have some function $V(x)=x+log(c)$, where $x$ is a continuous random variable and $c$ a constant bounded on $[0,1]$. I have some queries regarding the following:
i) May the above function ...
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1answer
24 views
Distribution of $F_{Z}(z)$ where $Z = min\{X,Y\}$
I'm having trouble understanding a step in my teacher's explanation.
$F_{Z}(z) =\mathbb{P}(Z\leq z) = \mathbb{P}(min\{X,Y\} \leq z) = 1 - \mathbb{P}(max\{X,Y\} > z) = 1 - \mathbb{P}(X>z, Y>z)...
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0answers
29 views
Minimizing Mean Squared Error Of Prediction From Adding Two Random Variables
First, assume you have variables $x$ and $y$. These variables are combined together to form a variable $t$ in the following way:
$$t=ax + by$$
Where a and b are weights, and t is essentially a ...
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0answers
41 views
Map $\operatorname{sinc}$ onto a random series of integers
Assume a real-valued function $f$ that yields $1$ for random integer values of $x$, and $0$ else. Let these instances of $f(x)=1$ be located at $x=\{n_j,n_{j+1},n_{j+2}...\}$.
I want to create a new (...
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1answer
65 views
Bounding the variance of the maximum of convex functions of a random variable
Consider a random variable $X$ with density $f$, and a finite set of convex and increasing functions $Y_i(x)$. I am interested in bounding the variance of the max of the $Y_i$'s.
I have the following ...
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0answers
47 views
Wide Sense Stationary Random Process
For the first part, I think I have to prove the 2 WSS conditions which are constant mean as well as the covariance sequence of C[k1,k2] = g(|k1 - k2|), where g is some function.
Im not really sure ...
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0answers
29 views
Applications of random homeomorphisms
I was reading A Gerald's "Integral, Probability and Fractal Measure" and I found the concept of $\bf{random}$ ${\bf homeomorphism}$ and I was wondering about the following:
What kind of applications ...
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1answer
40 views
Suitable distribution to make a problem tractable
Consider a set of random variables $x_i$ for $i\in\{1,\dots,n\}$. They are all drawn in an iid way from some distribution $F$. Now consider the function
$$
Y(x)=A\log\left(\sum_{i}B_{i}x_{i}^{a}\...
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1answer
53 views
Explanation of $\lim\sup$ of a sequence of random variables in measure theory
The definition I have been given of the $\limsup\limits_{n \to \infty} Y_n$ where the $Y_n$ are random variables is that it is another random variable defined as $(\limsup\limits_{n \to \infty} Y_n)(\...
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1answer
60 views
About sum of random variables with specific PDF
Given two independent random variables X and Y, where CDF and PDF for X are:
\begin{equation}
F_X(x) = \Biggl\{
\begin{array}{ll}
0 & \mbox{if $x \le 0$}\\
...
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1answer
29 views
What does this vector function notation mean?
Under "Functions of Random Vectors: The Method of Transformations"
https://www.probabilitycourse.com/chapter6/6_1_5_random_vectors.php
The notation says let $G:\Bbb R^n \rightarrow \Bbb R^n$ be a ...
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0answers
42 views
pdf of a function of a normal random variable
Let $f$ be a function $f : x \mapsto y$ where $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$; $m \geq n$. $f$ is not invertible. I have a random variable $X$ s.t. $X \sim \mathcal{N}(\mu, \Sigma)$ ...
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1answer
39 views
using monte carlo to determine uncertainty.
In my thesis there are some uncertainties (for example in geometry: the diameter of cylinder, the height ,.. or the temperature of inlet fluid ) and I want to know what is the effect of them in my ...
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0answers
31 views
Random measures by random fields
Given a probability space $(\Omega,\mathcal{A},\mathbb{P})$, we have a random field $\{X_t\}_{t \in T}$, $T\subset S_1\times S_2$, for a measurable space $(S_1 \times S_2,\mathcal{A}_1\times\mathcal{A}...
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2answers
64 views
How to efficiently sample data from a known cumulative distribution of a *function* of a variable?
Problem
I need to sample the diameter of some spheres starting from a given fractional volume distribution, which represents the volume percentage as a function of diameter $d$ and is given as
$$ \...
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1answer
35 views
Calculating the expected value and variance of a sum of two different distributions
I have the following expression:
$$
Z = G - N + E
$$
where
$$
G \sim \mathrm{Gamma}(k,\frac{\sigma^2}{k}) \\
N \sim \mathcal{N}(0,\dfrac{\alpha{}\sigma^2}{k^{2}}) \\
E \in \mathbb{R}_{+}
$$
This ...
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1answer
136 views
Jensen's inequality for random functions in a Banach space
Suppose I have a Banach space of functions over $\mathbb{R}$ with norm $||\cdot||$. Suppose that $f$ is a random function that takes values in this space such that $||f||\leq 1$. Suppose that for all $...
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0answers
31 views
Random process. Co-distribution of process and its derivative.
As part of a solution of my problem I need to find $\rho(\xi,\xi')$,
where $\xi=\frac{1}{(1+(x')^2)^\frac32}$ and x is a gaussian stationary process.
So I`ve found $\rho(\xi)$ , I found $\xi'$ and I ...
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1answer
109 views
In algebra of random variables is $X + X \neq 2X$?
In the algebra of random variables, is the following identity correct?
$$X + X \neq 2X$$
The underlying source of this questions is a mathematical text that considers a MLE estimator for mean of a ...
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0answers
52 views
A minimization chance constrained optimization problem
$\mathbf{Q}:$ Suppose we have the following chance constrained optimization problem:
\begin{aligned}
& \text{minimize}
& & x_1 +x_2\\
& \text{s.t.}
...
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0answers
222 views
What is the PDF of this scaled non-central chi-squared random variable and what is the result of its integral?
Consider a random variable that has a scaled non-central chi-squared
distribution
\begin{eqnarray*}
L & = & a\chi_{1}^{2}(b^{2}),
\end{eqnarray*}
where $a$ is a positive scalar and $\chi_{1}^{...
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2answers
123 views
what are the characteristics of this random variable?
Consider independent sequences of independent and identically distributed
(i.i.d.) zero-mean Gaussian random scalars $r_{i}\sim\mathcal{N}(\theta,\sigma^{2})$
and $\rho\sim\mathcal{N}(0,a^{2})$. ...
2
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1answer
92 views
Integral with respect to random measure is measurable
Let $(\Omega, \mathcal{F})$ be a measurable space and $P$ be a random, $\mathcal{G}$-measurable finite measure on $(\Omega, \mathcal{F})$, with $\mathcal{G} \subseteq \mathcal{F}$.
Is the following ...
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2answers
25 views
Will operations on a purely random function also be random?
Let's assume that a function random(x,y) gives a completely random number as output which will be in between x and y (not a pseudo random function) such that ...
1
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1answer
73 views
Linear combination of non-identically distributed, independent exponential random variables
I am working on the following homework assignment:
Under the assumptions of the Normal Simple Linear Regression model, $Y_i|X_i \sim N(\beta_0 + \beta_1 X_i, \sigma^2)$. Consider the model where $...
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1answer
28 views
Sequence of Random Variables: Approximation to the probability of an event
i'd like some help with the following problem, please.
Let there be a sequence of random variables
$ X_{n}$ such that $n\geq 0$.
If $X_{n} \sim Beta (n,1)$ such that Beta distribution is ...
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3answers
316 views
Projections of uniformly distributed $\mathbb{R}^3$ unit vector have uniform distribution
My question revolves around the following property:
Let ${\bf u} \in \mathbb{R}^3$ be a random vector with uniform distribution on the three-dimensional unit sphere. Then the projection on any given ...
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2answers
25 views
Order Statistics and Functions of Random Variables
I'd like some help with following problem, please.
$X \sim Bin(5; 0,2) \\
Y = \frac{1}{2}(Max(X,4) + Min(X,2))$
I need to find Y's density, but i don't really know how to apply Jacobian method or ...
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0answers
53 views
$X,Y \sim Exp(1), X \perp Y$, joint density of $V = X, U = X/Y$
I'd like to check if my results are correct. Could you guys please verify them?
$X,Y \sim \; Exp(1), X \perp Y$
Find the joint density of $U = X, V = X/Y$:
\begin{align}
V = X/Y & \rightarrow ...
0
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0answers
61 views
Does MCMC method can be used to calculate the mean and variance of the distribution of random variable functions?
I am not professional in Probability & Statisticsin, in order to clearly describe my problem, please be patient of the long introduction.THANKS!
Background of my question
Assume I have several ...
