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Questions tagged [random-functions]

Functions of random variables.

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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 ...
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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 ...
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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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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. ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 $\rho\sim\mathcal{N}(0,a^{2})$. ...
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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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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 ...
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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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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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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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$X,Y \sim Exp(1), X \perp Y$, 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 ...
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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 ...